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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = 1;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1;<br/>
<a NAME="E1:58"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="funct_7.html#K1">In</a> 0,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> 1</span>)</span>
 
;<br/>
<a NAME="E3:58_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:58_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> st <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E42</font></i>:  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<a NAME="E3:58"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:58"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:58"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:58"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:58"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<a NAME="E1:58_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:58_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:58"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:58"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:58_9"/>
( <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:58_9"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:58"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:58"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
0
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T13">GR_CY_1:13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E12:58"/><i><font color="Green">E52</font></i>: 
 <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>take </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E1:58_12"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E8:58"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:58"/>
 <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="vectsp_1.html#V8">left_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a><br/>

<b>thus </b><a NAME="E1:58_13"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="int_3.html#E9:58"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:58"/>
(  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="vectsp_1.html#V10">degenerated</a> &amp;  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is   <a href="funcsdom.html#NM1">Ring</a> &amp;  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="vectsp_1.html#V9">Field-like</a> &amp;  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="group_1.html#V2">unital</a> &amp;  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="vectsp_1.html#V7">distributive</a> &amp;  <a href="int_3.html#K9" target="_self">INT.Ring</a> 1 is <a href="group_1.html#V7">commutative</a> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:58"><i><font color="Green">E41</font></i></a>, <a class="txt" href="int_3.html#E2:58"><i><font color="Green">E42</font></i></a>, <a class="txt" href="int_3.html#E3:58"><i><font color="Green">E43</font></i></a>, <a class="txt" href="int_3.html#E4:58"><i><font color="Green">E44</font></i></a>, <a class="txt" href="int_3.html#E5:58"><i><font color="Green">E45</font></i></a>, <a class="txt" href="int_3.html#E6:58"><i><font color="Green">E46</font></i></a>, <a class="txt" href="int_3.html#E7:58"><i><font color="Green">E47</font></i></a>, <a class="txt" href="int_3.html#E9:58"><i><font color="Green">E49</font></i></a>, <a class="txt" href="int_3.html#E10:58"><i><font color="Green">E50</font></i></a>, <a class="txt" href="int_3.html#E11:58"><i><font color="Green">E51</font></i></a>, <a class="txt" href="int_3.html#E12:58"><i><font color="Green">E52</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a>, <a class="ref" href="group_1.html#D2">GROUP_1:def 2</a>, <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a>, <a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>


</div>
