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

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







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</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="E1:67_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</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>7</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="E3:67_1"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:67_1"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:67_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T67" target="_self">Th67</a></i>;<br/><a NAME="E6:67_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <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"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:67_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="group_4.html#E4:67_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="group_4.html#E1:67_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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="txt" href="group_4.html#E1:67_1_1_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</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>11</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="E5:67_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <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>11</sub></font>
 <b>and </b><br/><a NAME="E6:67_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <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>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:67_1_1_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>9</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>10</sub></font> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E2:67_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="group_4.html#T67" target="_self">Th67</a></i>;<br/><a NAME="E8:67_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_4.html#E95"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E9:67_1_1_1_1_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#R1">=</a>  <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>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><a NAME="E10:67_1_1_1_1_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E11:67_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><a NAME="E12:67_1_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <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>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E6:67_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:67_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E5:67_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="group_4.html#E7:67_1_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="group_4.html#T67" target="_self">Th67</a></i>;<br/><a NAME="E14:67_1_1_1_1_1"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E15:67_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</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"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</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="67_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>4</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_4.html#E95"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E3:67_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/><a NAME="E4:67_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E5:67_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E6:67_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</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>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</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"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="group_4.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:67_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</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#K5" target="_self">gr</a> <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>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E8:67_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</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"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:67_1_1"/>
<font color="Maroon">c<sub>8</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"><span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:67_1"/><b>then </b>
<font color="Maroon">c<sub>5</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"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><a href="group_4.html#K5" target="_self">gr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_4.html#E3:67_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="group_4.html#E5:67_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="group_4.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E8:67_1"/><b>then </b>
<font color="Maroon">c<sub>5</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"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E9:67_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:67"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K8" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T67">GROUP_2:67</a></i>;<br/>


</div>
