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

<b>set </b><font color="Maroon" title="c2">n</font> = 1;<br/>
<b>set </b><font color="Maroon" title="c3">R</font> =  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1;<br/>
<a NAME="E1:51"/><i><font color="Green" title="E34">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> st <font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E1:51_1"/>
<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> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:51_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:51_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> st <font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E35">A2</font></i>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>
<a NAME="E3:51"/><i><font color="Green" title="E36">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">b</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/><i><font color="Green" title="E37">A4</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:51"/><i><font color="Green" title="E38">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">c</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:51"/><i><font color="Green" title="E39">A6</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:51"/><i><font color="Green" title="E40">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<a NAME="E1:51_7"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:51_7"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:51"/><i><font color="Green" title="E41">A8</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:51"/><i><font color="Green" title="E42">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds <br/> ( <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<i><font color="Green" title="E43">A10</font></i>: <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:51_9"/>
( <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:51_9"><i><font color="Green" title="E43">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:51"/><i><font color="Green" title="E43">A11</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:51"/><i><font color="Green" title="E44">A12</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13" title="GR_CY_1:th.13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E12:51"/><i><font color="Green" title="E45">A13</font></i>: 
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>take </b>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">v</font>
;<br/>

<b>thus </b><a NAME="E1:51_12"/>
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">v</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> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E8:51"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:51"/>
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a><br/>

<b>thus </b><a NAME="E1:51_13"/>
<font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="int_3.html#E9:51"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:51_13"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="int_3.html#E9:51"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:51"/>
(  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> &amp;  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> &amp;  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">almost_left_invertible</a> &amp;  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> &amp;  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> &amp;  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> 1 is <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:51"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="int_3.html#E2:51"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="int_3.html#E3:51"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="int_3.html#E4:51"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="int_3.html#E5:51"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="int_3.html#E6:51"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="int_3.html#E7:51"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="int_3.html#E9:51"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="int_3.html#E10:51"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="int_3.html#E11:51"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="int_3.html#E12:51"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="group_1.html#D2" title="GROUP_1:def.2">GROUP_1:def 2</a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0:def 8</a>, <a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a>, <a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/>


</div>
