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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> or for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:19_1_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:19_1_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E3:19_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="group_8.html#T17" target="_self">Th17</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:19_1_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E3:19_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="group_8.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:19_1_2"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:19_1_2_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="group_8.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E4:19_1_2_1_1"/>
ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <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="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E2:19_1_2_1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E3:19_1_2_1_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="group_8.html#E2:19_1_2_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E7:19_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<a NAME="E5:19_1_2_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
;<br/>
<b>take </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>;<br/>

<a NAME="E6:19_1_2_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:19_1_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="group_8.html#E5:19_1_2_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E7:19_1_2_1_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_1_1_1_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E3:19_1_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E2:19_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E7:19_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:19_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_1_1_1_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E7:19_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E2:19_1_2_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E3:19_1_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19_1_2_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E6:19_1_2_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_8.html#E7:19_1_2_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="group_8.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:19_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_2_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:19_1_2_2_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="group_8.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E4:19_1_2_2_1"/>
ex <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 <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <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="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E2:19_1_2_2_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E3:19_1_2_2_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<b>then </b><i><font color="Green">E19</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="group_8.html#E2:19_1_2_2_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E7:19_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<b>take </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>;<br/>

<a NAME="E5:19_1_2_2_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:19_1_2_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="group_8.html#E4:19_1_2_2_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:19_1_2_2_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_2_1_1_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E7:19_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="group_8.html#E3:19_1_2_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E2:19_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:19_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_2_1_1_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E7:19_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E2:19_1_2_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E3:19_1_2_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:19_1_2_2_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E5:19_1_2_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_8.html#E6:19_1_2_2_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_1_2_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="group_8.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:19_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> or for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="group_8.html#E8:19_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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