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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="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>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</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> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:5"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,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_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/>
<a NAME="E5:5"/><i><font color="Green">E4</font></i>: 
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>
 
;<br/>
<a NAME="E6:5"/><i><font color="Green">E5</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E7:5"/><i><font color="Green">E6</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E7</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:5"><i><font color="Green">E3</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <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_2.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:5"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E6:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="group_2.html#T98">GROUP_2:98</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>
<a NAME="E9:5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</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_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<b>hence </b><a NAME="E2:5_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</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_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:5"><i><font color="Green">E3</font></i></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>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,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_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="grsolv_1.html#E8:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:5"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><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> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>

<a NAME="E1:5_3"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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#K9">/\</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>by </b><i><a class="ref" href="group_2.html#D10">GROUP_2:def 10</a></i>;<br/>
<a NAME="E2:5_3"/><b>then </b><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:5_3"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E2:5_3"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E5:5_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T106">GROUP_2:106</a></i>;<br/>
<a NAME="E6:5_3"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/>
<a NAME="E7:5_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:5_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:5_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E3:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="grsolv_1.html#E6:5_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<i><font color="Green">E8</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> = 
 <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_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:5"><i><font color="Green">E4</font></i></a>, <a class="txt" href="grsolv_1.html#E6:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="group_2.html#T98">GROUP_2:98</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>
<a NAME="E14:5"/><i><font color="Green">E9</font></i>: 
 <a href="group_6.html#K11">Ker</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="group_6.html#K11">Ker</a> <font color="Maroon">c<sub>9</sub></font> as    <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_2.html#T65">GROUP_2:65</a></i>;<br/>
<a NAME="E2:5_5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E1:5_5_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_5_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:5_5_1_1"/><b>then </b>
<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_6.html#K11">Ker</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E4:5_5_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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#K9">/\</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>by </b><i><a class="ref" href="group_2.html#D10">GROUP_2:def 10</a></i>;<br/>
<a NAME="E5:5_5_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:5_5_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E7:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:5_5_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E5:5_5_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E8:5_5_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</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> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E5:5_5_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<i><font color="Green">E14</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="grsolv_1.html#E1:5_5_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="group_6.html#T50">GROUP_6:50</a></i>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#T29">GROUP_6:29</a></i>
;<br/>
<a NAME="E11:5_5_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="grsolv_1.html#E8:5_5_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="group_6.html#D9">GROUP_6:def 9</a></i>;<br/>
<a NAME="E12:5_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E10:5_5_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>;<br/>
<b>hence </b><a NAME="E13:5_5_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="grsolv_1.html#E6:5_5_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>


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

<b>thus </b><a NAME="E2:5_5_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_5_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:5_5_1_2"/><b>then </b>
<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"><font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</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="E3:5_5_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>11</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"><font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<a NAME="E4:5_5_1_2"/><i><font color="Green">E11</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</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> ;<br/>
<i><font color="Green">E12</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> = 
<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#K9">/\</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#D10">GROUP_2:def 10</a></i>
;<br/>
<a NAME="E7:5_5_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:5_5_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:5_5_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font></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"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E13:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="grsolv_1.html#E4:5_5_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="grsolv_1.html#E6:5_5_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="grsolv_1.html#E3:5_5_1_2"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E10:5_5_1_2"/>
<font color="Maroon">c<sub>13</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>
 
<b>by </b><i><a class="txt" href="grsolv_1.html#E7:5_5_1_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E11:5_5_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E12:5_5_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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#K9">/\</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>by </b><i><a class="ref" href="group_2.html#D10">GROUP_2:def 10</a></i>;<br/>
<a NAME="E13:5_5_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:5_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E15:5_5_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10">nat_hom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="grsolv_1.html#E13:5_5_1_2"><i><font color="Green">E15</font></i></a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E3:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="grsolv_1.html#E15:5_5_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="group_6.html#D9">GROUP_6:def 9</a></i>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="grsolv_1.html#E11:5_5_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T29">GROUP_6:29</a></i>
;<br/>
<b>hence </b><a NAME="E18:5_5_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="group_6.html#T50">GROUP_6:50</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_5"/>
 <a href="group_6.html#K11">Ker</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#D6">GROUP_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="group_6.html#K11">Ker</a> <font color="Maroon">c<sub>9</sub></font> as  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<a NAME="E16:5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>10</sub></font>, <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_6.html#R2">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T90">GROUP_6:90</a></i>;<br/>
<b>hence </b><a NAME="E17:5"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2">/\</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K6">./.</a> <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_6.html#K6">./.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="group_6.html#R1">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="grsolv_1.html#E14:5"><i><font color="Green">E9</font></i></a></i>;<br/>


</div>
