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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">c<sub>2</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="1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>hence </b> <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="txt" href="rat_1.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="arytm_3.html#D10">ARYTM_3:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xcmplx_1.html#T49">XCMPLX_1:49</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>then consider </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>, <font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:1_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E6:1_1_2"/><i><font color="Green">E6</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E7:1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E4:1_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E8:1_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E4:1_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E9:1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E5:1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E10:1_1_2"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E5:1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E11:1_1_2"/>
 <a href="arytm_0.html#K1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/><a NAME="E12:1_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="arytm_0.html#K3">opp</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/><a NAME="E13:1_1_2"/>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E6:1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><b>then </b><i><font color="Green">E11</font></i>: 1 = 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E6:1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E7:1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="rat_1.html#E12:1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E16:1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E17:1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E18:1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E19:1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E16:1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E20:1_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E16:1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E21:1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E17:1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E22:1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E17:1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E23:1_1_2"/>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E18:1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><b>then </b><i><font color="Green">E18</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> = 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E10:1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rat_1.html#E18:1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rat_1.html#E21:1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rat_1.html#E22:1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,0
<b>by </b><i><a class="txt" href="rat_1.html#E12:1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><a NAME="E25:1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E20:1_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E26:1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E27:1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green">E8</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E30:1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="arytm_3.html#T60">ARYTM_3:60</a></i>;<br/><a NAME="E31:1_1_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a></i>;<br/><a NAME="E33:1_1_2"/><i><font color="Green">E22</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="arytm_2.html#T22">ARYTM_2:22</a></i>;<br/><a NAME="E34:1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1">ARYTM_0:1</a></i>;<br/><a NAME="E36:1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><a NAME="E37:1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rat_1.html#E14:1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rat_1.html#E30:1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rat_1.html#E33:1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rat_1.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="arytm_0.html#T20">ARYTM_0:20</a></i>;<br/><a NAME="E38:1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E40:1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E41:1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E42:1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E24:1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rat_1.html#E26:1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a>, <a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E44:1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E45:1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E46:1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E25:1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rat_1.html#E37:1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rat_1.html#E40:1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rat_1.html#E41:1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rat_1.html#E42:1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="arytm_2.html#T22">ARYTM_2:22</a>, <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E47:1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E49:1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green">E8</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E51:1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E52:1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>and </b><br/><a NAME="E53:1_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="rat_1.html#E38:1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a>, <a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a>, <a class="ref" href="arytm_2.html#T2">ARYTM_2:2</a></i>;<br/><a NAME="E54:1_1_2"/><i><font color="Green">E35</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rat_1.html#E37:1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rat_1.html#E47:1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rat_1.html#E51:1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rat_1.html#E52:1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="arytm_2.html#T22">ARYTM_2:22</a>, <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><font color="Maroon">c<sub>22</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>2</sub></font>,<a href="arytm_3.html#K14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T46">ARYTM_3:46</a></i>
<br/>.= 
 <a href="arytm_3.html#NK11" target="_self">quotient</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="arytm_3.html#K3">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="arytm_3.html#K3">*^</a> <a href="arytm_3.html#K14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T55">ARYTM_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T55">ARYTM_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="arytm_3.html#T47">ARYTM_3:47</a></i>
<br/>.= 
 <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="arytm_3.html#T46">ARYTM_3:46</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="txt" href="rat_1.html#E20:1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rat_1.html#E40:1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rat_1.html#E44:1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rat_1.html#E14:1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rat_1.html#E41:1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rat_1.html#E45:1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="rat_1.html#E51:1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rat_1.html#E52:1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rat_1.html#E53:1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rat_1.html#E54:1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rat_1.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
;<br/><b>hence </b> <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rat_1.html#E46:1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="arytm_3.html#T62">ARYTM_3:62</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>

;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
