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

<span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="int_3.html#NM1" title="INT_3:NM.1">EuclidianRing</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">E</font> is   <a href="gcd_1.html#NM3" title="GCD_1:NM.3">gcdDomain</a></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">d</font> being    <a href="int_3.html#M1" title="INT_3:mode.1">DegreeFunction</a> of <font color="Maroon" title="c2">E</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b2">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">z</font> ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:29_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:29_1_1_1_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b2">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">z</font> ) )</span><br/><div class="add"><a NAME="E2:29_1_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>;<br/><a NAME="E3:29_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>;<br/><a NAME="E4:29_1_1_1_1"/>
for <font color="Olive" title="b1">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b1">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font>
 ;<br/><span class="kw">hence </span><a NAME="E5:29_1_1_1_1"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b2">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">z</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:29_1_1_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:29_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b2">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">z</font> ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> ;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b3">s</font>, <font color="Olive" title="b4">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span>  &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> );<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><a NAME="E3:29_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 ;<br/><a NAME="E4:29_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 ;<br/><a NAME="E5:29_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/><a NAME="E6:29_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E5:29_1_1_1_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>);<br/></span><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:29_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c8">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E10:29_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E11:29_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">and </span><br/><a NAME="E12:29_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  ex <font color="Olive" title="b2">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">z</font> where <font color="Olive" title="b3">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b4">s</font>, <font color="Olive" title="b5">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span>  &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">z</font> ) holds <br/><font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_1_1_1_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span> ;<br/><a NAME="E13:29_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b3">s</font>, <font color="Olive" title="b4">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span>  holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span>  implies <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_1_2_2"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E3:29_1_1_1_2_2"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font>
 <span class="kw">and </span><br/><a NAME="E4:29_1_1_1_2_2"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">u</font>, <font color="Maroon" title="c13">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E6:29_1_1_1_2_2"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E26">A11</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">q</font>, <font color="Maroon" title="c16">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E9:29_1_1_1_2_2"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">and </span><br/><a NAME="E10:29_1_1_1_2_2"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
( <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> or <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_1_1_1_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="int_3.html#D10" target="_self" title="INT_3:def.10">Def10</a></span>;<br/>
<a NAME="E11:29_1_1_1_2_2"/>
<font color="Maroon" title="c16">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c17">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E2:29_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">z1</font>
 <span class="kw">and </span><br/><a NAME="E3:29_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c17">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">s</font>, <font color="Maroon" title="c19">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E5:29_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c17">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a></span>;<br/>
<font color="Maroon" title="c14">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c16">r</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c16">r</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c16">r</font> = 
<font color="Maroon" title="c14">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E5:29_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></span>
<br/>.= 
<font color="Maroon" title="c14">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T45" title="RLVECT_1:th.45">RLVECT_1:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E6:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T51" title="GCD_1:th.51">GCD_1:51</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#T51" title="GCD_1:th.51">GCD_1:51</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c18">s</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c18">s</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c18">s</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c18">s</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">q</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">t</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:29_1_1_1_2_2_1"/>
<font color="Maroon" title="c16">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:29_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29_1_1_1_2"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E10:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a></span>;<br/>
<a NAME="E13:29_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1_1_1_2_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:29_1_1_1_2_2"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:29_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c8">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c8">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E2:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c8">g</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E4:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E29">A22</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">zz</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E6:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">zz</font>
 <span class="kw">and </span><br/><a NAME="E7:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c12">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c14">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E9:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c12">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E31">A24</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E11:29_1_1_1_2_3"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>;<br/>
<font color="Maroon" title="c8">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="txt" href="#E11:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="txt" href="#E6:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="txt" href="#E9:29_1_1_1_2_3"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c15">v</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c15">v</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:29_1_1_1_2_3"/>
<font color="Maroon" title="c10">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><a NAME="E16:29_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A27</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 ;<br/><a NAME="E17:29_1_1_1_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span>  holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b3">s</font>, <font color="Olive" title="b4">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span>  implies <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_1_2_5"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E3:29_1_1_1_2_5"/><span class="lab"><font color="Green" title="E28">A28</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font>
 <span class="kw">and </span><br/><a NAME="E4:29_1_1_1_2_5"/><span class="lab"><font color="Green" title="E29">A29</font></span>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E28">A28</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E7:29_1_1_1_2_5"/><span class="lab"><font color="Green" title="E30">A30</font></span>: 
<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E29">A29</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E9:29_1_1_1_2_5"/><span class="lab"><font color="Green" title="E31">A31</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z1</font>
 <span class="kw">and </span><br/><a NAME="E10:29_1_1_1_2_5"/><span class="lab"><font color="Green" title="E32">A32</font></span>: 
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Maroon" title="c14">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">s</font>, <font color="Maroon" title="c16">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E12:29_1_1_1_2_5"/><span class="lab"><font color="Green" title="E33">A33</font></span>: 
<font color="Maroon" title="c14">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E32">A32</font></span></a></span>;<br/>
<font color="Maroon" title="c12">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E28">A28</font></span></a>, <a class="txt" href="#E7:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E30">A30</font></span></a>, <a class="txt" href="#E9:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E31">A31</font></span></a>, <a class="txt" href="#E12:29_1_1_1_2_5"><span class="lab"><font color="Green" title="E33">A33</font></span></a>, <a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c15">s</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:29_1_1_1_2_5"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:29_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">y</font></span>)</span> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> :  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:29_1_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/><a NAME="E19:29_1_1_1_2"/>
( <font color="Maroon" title="c8">g</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">g</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_1_1_1_2_6"/>
(  ex <font color="Olive" title="b1">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">zz</font> &amp;  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> ) &amp;  ex <font color="Olive" title="b1">zzz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">zzz</font> &amp;  ex <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <font color="Olive" title="b1">zzz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">g</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E16:29_1_1_1_2"><span class="lab"><font color="Green" title="E27">A27</font></span></a>, <a class="txt" href="#E18:29_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_1_1_1_2_6"/>
( <font color="Maroon" title="c8">g</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">g</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E20:29_1_1_1_2"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b2">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">z</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_1_1_1_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:29_1"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b2">zz</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b2">zz</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">z</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:29"/>
<font color="Maroon" title="c2">E</font> is   <a href="gcd_1.html#NM3" title="GCD_1:NM.3">gcdDomain</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D11" title="GCD_1:def.11">GCD_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
