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

<b>let </b><font color="Maroon">E</font> be   <a href="int_3.html#NM1">EuclidianRing</a>;<br/>

<b>consider </b><font color="Maroon">d</font> being    <a href="int_3.html#M1" target="_self">DegreeFunction</a> of <font color="Maroon">E</font>;<br/>
<a NAME="E2:34"/>
<font color="Maroon">E</font> is <a href="gcd_1.html#V3">gcd-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_1"><b>case </b></a><a NAME="E1:34_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><a NAME="E3:34_1_1_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="gcd_1.html#R2">divides</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/><a NAME="E4:34_1_1_1_1_1"/>
for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b><a NAME="E5:34_1_1_1_1_1"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Olive">z</font> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_2"><b>case </b></a><a NAME="E1:34_1_1_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span> ;<br/><a NAME="E2:34_1_1_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">E</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E3:34_1_1_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  &amp; <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> );<br/><a NAME="E3:34_1_1_1_1_2"/><i><font color="Green">E40</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_1_1_2_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
;<br/>
<b>hence </b><a NAME="E2:34_1_1_1_1_2_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i><a class="txt" href="int_3.html#E2:34_1_1_1_1_2"><i><font color="Green">E24</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:34_1_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a>);<br/></i><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:34_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 ;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E8:34_1_1_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  &amp; <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  &amp; <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span> ;<br/><a NAME="E9:34_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_1_1_2_3"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_1_1_2_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:34_1_1_1_1_2_3_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp;  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> )
 ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E6:34_1_1_1_1_2_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E8:34_1_1_1_1_2_3_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r</font> &amp; ( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font> or <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:34_1_1_1_1_2_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_3_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E53</font></i>: <font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
<font color="Maroon">r</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>
<b>consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:34_1_1_1_1_2_3_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp;  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:34_1_1_1_1_2_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/>
<font color="Maroon">r</font> = 
<font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="int_3.html#E8"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T45">RLVECT_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T51">GCD_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gcd_1.html#T51">GCD_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">t</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">t</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E7:34_1_1_1_1_2_3_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:34_1_1_1_1_2_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E11:34_1_1_1_1_2_3_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:34_1_1_1_1_2_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_1_1_1_1_2_3"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span>  holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_1_1_2_3_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:34_1_1_1_1_2_3_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp;  ex <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E6:34_1_1_1_1_2_3_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E8:34_1_1_1_1_2_3_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp;  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E10:34_1_1_1_1_2_3_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a>, , <a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">t</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E12:34_1_1_1_1_2_3_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34_1_1_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> :  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:34_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">g</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">zz</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E2:34_1_1_1_1_2_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">zz</font> &amp;  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">zzz</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E4:34_1_1_1_1_2_4"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">zzz</font> &amp;  ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">zzz</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:34_1_1_1_1_2_4"/>
( <font color="Maroon">g</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:34_1_1_1_1_2"/>
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_1_1_2_5"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:34_1_1_1_1_2_5"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:34_1_1_1_1_2_5"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">zz</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E7:34_1_1_1_1_2_5"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">zz</font> &amp;  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>, <font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E9:34_1_1_1_1_2_5"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/>
<font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a>, , , <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E11:34_1_1_1_1_2_5"/>
<font color="Maroon">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1">GCD_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:34_1_1_1_1_2"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:34_1_1"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">zz</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">x</font> &amp; <font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Maroon">y</font> holds <br/><font color="Olive">zz</font> <a href="gcd_1.html#R2">divides</a> <font color="Olive">z</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon">E</font> is <a href="gcd_1.html#V3">gcd-like</a>
 <b>by </b><i><a class="ref" href="gcd_1.html#D11">GCD_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
<font color="Maroon">E</font> is   <a href="gcd_1.html#NM3">gcdDomain</a>
 ;<br/>


</div>
