<?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>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:52"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:52_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> 
 ;<br/>

<a NAME="E3:52_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E1:52_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E6:52_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="group_4.html#V1" target="_self">generating</a>
 
<b>by </b><i><a class="txt" href="group_4.html#E2:52_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:52_1"/><i><font color="Green">E37</font></i>: 
 <a href="group_4.html#K5" target="_self">gr</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E9:52_1"/><i><font color="Green">E38</font></i>: 
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#D6" target="_self">Def6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:52_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E12:52_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="group_4.html#S2" target="_self">GROUP_4:sch 2</a>();<br/></i>
<a NAME="E13:52_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E14:52_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:52_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:52_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:52_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:52_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E6:52_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:52_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="group_4.html#E3:52_1_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E6:52_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="group_4.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:52_1_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E3:52_1_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:52_1_1_1_1"/><b>then </b><i><font color="Green">E48</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>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_4.html#E4:52_1_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="group_4.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="group_4.html#E8:52_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_4.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:52_1_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="group_4.html#E1:52_1_1_1_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E6:52_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_4.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E7:52_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:52_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:52_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="group_4.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E9:52_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="group_2.html#T71">GROUP_2:71</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E14:52_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <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> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> );<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E1:52_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E2:52_1_2"/>
<font color="Maroon">c<sub>9</sub></font> is  <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>
 <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:52_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <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> <b>by </b><i><a class="txt" href="group_4.html#E17:52_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E2:52_1_3"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T63">GROUP_2:63</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:52_1"/><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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/>
<a NAME="E20:52_1"/><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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>given </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being  <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><b> such that </b><a NAME="E1:52_1_4"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being  <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><b> such that </b><a NAME="E2:52_1_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>

<a NAME="E3:52_1_4"/>
<font color="Maroon">c<sub>12</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E1:52_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_4.html#E2:52_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="group_2.html#T65">GROUP_2:65</a></i>;<br/>
<b>hence </b><a NAME="E4:52_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_4.html#E2:52_1_4"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:52_1"/><i><font color="Green">E46</font></i>: 
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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_1_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:52_1_5"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_1"><b>now </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="E1:52_1_5_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/><a NAME="E2:52_1_5_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</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> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>hence </b><a NAME="E3:52_1_5_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_5_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>assume </b><a NAME="E3:52_1_5_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>hence </b><a NAME="E4:52_1_5_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font>]
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5_2_1_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E3:52_1_5_2_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="group_4.html#E1:52_1_5_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <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> <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/><a NAME="E5:52_1_5_2_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5_2_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:52_1_5_2_1_2"/><b>then </b>
(  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E3:52_1_5_2_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:52_1_5_2_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:52_1_5_2_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="orders_1.html#T91">ORDERS_1:91</a></i>;<br/><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:52_1_5_2_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E2:52_1_5_2_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_1_5_2_1_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:52_1_5_2_1_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5_2_1_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:52_1_5_2_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E8:52_1_5_2_1_2_1"/><i><font color="Green">E60</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:52_1_5_2_1_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_5_2_1_2_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:52_1_5_2_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E13:52_1_5_2_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E2:52_1_5_2_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:52_1_5_2_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E16:52_1_5_2_1_2_1"/><i><font color="Green">E65</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E13:52_1_5_2_1_2_1"><i><font color="Green">E63</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E18:52_1_5_2_1_2_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E16:52_1_5_2_1_2_1"><i><font color="Green">E65</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_5_2_1_2_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>21</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>18</sub></font>] )
 </span><b>by </b><i><a class="txt" href="group_4.html#E1:52_1_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="group_4.html#E2:52_1_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green">E66</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>21</sub></font>]
 ;<br/><div class="add"><a NAME="E2:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E3:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E4:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E5:52_1_5_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E12:52_1_5_2_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E15:52_1_5_2_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="group_2.html#T89">GROUP_2:89</a></i>;<br/><b>hence </b><a NAME="E6:52_1_5_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E13:52_1_5_2_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="group_4.html#E15:52_1_5_2_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_5_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_5_2_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>18</sub></font>]
 ;<br/><div class="add"><a NAME="E2:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E3:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E4:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E5:52_1_5_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="group_4.html#E12:52_1_5_2_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="group_4.html#E15:52_1_5_2_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="group_4.html#E18:52_1_5_2_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="group_2.html#T89">GROUP_2:89</a></i>;<br/><b>hence </b><a NAME="E6:52_1_5_2_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:52_1_5_2_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:52_1_5_2_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_5_2_1_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E4:52_1_5_2_1_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:52_1_5_2_1_2_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E7:52_1_5_2_1_2_2"/><i><font color="Green">E58</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_2"><i><font color="Green">E56</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:52_1_5_2_1_2_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_5_2_1_2_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E10:52_1_5_2_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E3:52_1_5_2_1_2_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="group_4.html#E6:52_1_5_2_1_2_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_4.html#E9:52_1_5_2_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="group_2.html#T90">GROUP_2:90</a></i>;<br/><b>hence </b><a NAME="E11:52_1_5_2_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="group_4.html#E6:52_1_5_2_1_2_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_4.html#E9:52_1_5_2_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being  <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><b> such that </b><br/><a NAME="E12:52_1_5_2_1_2"/><i><font color="Green">E55</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_5_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="group_4.html#E9:52_1_5_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_5_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_5_2_1_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E2:52_1_5_2_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1_5_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_1_5_2_1_2_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E5:52_1_5_2_1_2_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:52_1_5_2_1_2_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E8:52_1_5_2_1_2_3"/><i><font color="Green">E60</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_5_2_1_2_3"><i><font color="Green">E58</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <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><b> such that </b><br/><a NAME="E10:52_1_5_2_1_2_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1_5_2_1_2_3"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E11:52_1_5_2_1_2_3"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E1:52_1_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E12:52_1_5_2_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_5_2_1_2_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_4.html#E10:52_1_5_2_1_2_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E14:52_1_5_2_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_5_2_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="group_4.html#E12:52_1_5_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="group_4.html#E13:52_1_5_2_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>assume </b><a NAME="E16:52_1_5_2_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E17:52_1_5_2_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E11:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <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> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>thus </b><a NAME="E19:52_1_5_2_1_2"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><a NAME="E20:52_1_5_2_1_2"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font>
 ;<br/><a NAME="E21:52_1_5_2_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_4.html#E16:52_1_5_2_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E22:52_1_5_2_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E23:52_1_5_2_1_2"/>
<font color="Maroon">c<sub>19</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E12:52_1_5_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="group_2.html#T66">GROUP_2:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:52_1_5"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E23:52_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S1">ORDERS_1:sch 1</a>(<a class="txt" href="group_4.html#E18:52_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="group_4.html#E19:52_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="group_4.html#E20:52_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="group_4.html#E21:52_1"><i><font color="Green">E46</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <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> <b>by </b><i><a class="txt" href="group_4.html#E17:52_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E25:52_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:52_1_6"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:52_1_6"/>
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>10</sub></font> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_4.html#D7" target="_self">GROUP_4:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>11</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/>

<b>assume </b><b>that </b><br/><a NAME="E3:52_1_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>10</sub></font> #)
 <b>and </b><br/><a NAME="E4:52_1_6"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:52_1_6"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> #)
 ;<br/>

<a NAME="E6:52_1_6"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E4:52_1_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E7:52_1_6"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_6_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:52_1_6_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:52_1_6_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:52_1_6_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E7:52_1_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E5:52_1_6_1"/>
<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>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_6_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_6_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_6_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_6_1_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_6_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_6_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:52_1_6_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_6_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:52_1_6_1_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:52_1_6_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_1_6_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:52_1_6_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_1_6_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:52_1_6_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_6_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:52_1_6_1"/><b>then </b>
 <a href="group_4.html#K5" target="_self">gr</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E8:52_1_6_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E8:52_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="group_4.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E9:52_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E5:52_1_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_4.html#E7:52_1_6"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E10:52_1_6"/>
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 
<b>by </b><i><a class="txt" href="group_4.html#E23:52_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="group_4.html#E3:52_1_6"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:52_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E4:52_1_6"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:52_1"/><b>then </b>
 <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="group_4.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E27:52_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E28:52_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="group_4.html#E1:52_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="group_4.html#E12:52_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E29:52_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:52"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : not <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V1" target="_self">generating</a> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</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="E4:52"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:52"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> is <a href="group_4.html#V1" target="_self">generating</a>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_2_1"/>
( for <font color="Olive">b<sub>1</sub></font> being <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> holds not <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a> or ex <font color="Olive">b<sub>1</sub></font> being <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> st <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1"><b>suppose </b></a><a NAME="E1:52_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being <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> holds not <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div class="add"><a NAME="E2:52_2_1_1"/><b>then </b>
 <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E3:52_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_2"><b>suppose </b></a><a NAME="E1:52_2_1_2"/><i><font color="Green">E37</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <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> st <font color="Olive">b<sub>1</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><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/><b>assume </b><a NAME="E1:52_2_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="group_4.html#V2" target="_self">maximal</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2_1_2_1_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:52_2_1_2_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="group_4.html#T48" target="_self">Th48</a></i>;<br/><a NAME="E3:52_2_1_2_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:52_2_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</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"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:52_2_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:52_2_1_2_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> = 
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="group_4.html#E3:52_2_1_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
;<br/><a NAME="E5:52_2_1_2_1_1"/><b>then </b>
(  <a href="group_4.html#K5" target="_self">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="group_4.html#D7" target="_self">Def7</a>, <a class="ref" href="group_4.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E6:52_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_4.html#E5:52"><i><font color="Green">E36</font></i></a>, <a class="txt" href="group_4.html#E2:52_2_1_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="group_4.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_2_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E3:52_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E1:52_2_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="group_4.html#T52" target="_self">Th52</a></i>;<br/><b>hence </b><a NAME="E4:52_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E4:52"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:52"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" target="_self">Phi</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
