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

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Olive">b<sub>2</sub></font></span>)</span> );<br/>
<a NAME="E1:16_1_1"/><i><font color="Green">E14</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="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E1:16_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E2:16_1_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> )
 ;<br/>

<i><font color="Green">E17</font></i>: <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="ami_4.html#E1:16_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ami_4.html#E2:16_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>
;<br/>
<font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="ami_4.html#E1:16_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ami_4.html#E2:16_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>
;<br/>
<b>hence </b><a NAME="E5:16_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="ami_4.html#E1:16_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ami_4.html#E2:16_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ami_4.html#E3:16_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1">FUNCT_1:sch 1</a>(<a class="txt" href="ami_4.html#E1:16_1_1"><i><font color="Green">E14</font></i></a>);<br/></i>
<a NAME="E4:16_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:16_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> 
 <b>by </b><i><a class="txt" href="ami_4.html#E3:16_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:16_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_3"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E4:16_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ami_4.html#E3:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ami_4.html#E3:16_1_1_3"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> 
 <b>by </b><i><a class="ref" href="ami_3.html#T31">AMI_3:31</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> , <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a>  <b>by </b><i><a class="txt" href="ami_4.html#E4:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="ami_1.html#M1">FinPartState</a> of  <a href="ami_3.html#K1">SCM</a> ;<br/>

<b>thus </b><a NAME="E7:16_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="ami_4.html#E3:16_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E8:16_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 1</span>)</span> <a href="ami_3.html#K18">--&gt;</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> 0</span>)</span> <a href="ami_3.html#K17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> )
 ;<br/>

<a NAME="E9:16_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="ami_3.html#K1">SCM</a> 
 
<b>by </b><i><a class="ref" href="ami_3.html#T31">AMI_3:31</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="ami_4.html#E3:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ami_4.html#E8:16_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div>
