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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">N</font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<b>set </b><font color="Maroon">f</font> =  <a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">J</font> = <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> ;<br/>
<a NAME="E1:103"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B</font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T78">GROUP_4:78</a></i>;<br/>
<a NAME="E2:103"/><i><font color="Green">E46</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="group_2.html#D5">GROUP_2:def 5</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E3:103"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:103"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">N</font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T78">GROUP_4:78</a></i>;<br/>
<a NAME="E5:103"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">N</font> <a href="group_2.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:103"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:103"/>
 <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" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</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"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:103_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:103_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span>
 <b>and </b><br/><a NAME="E4:103_1"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i><a class="txt" href="group_6.html#E6:103"><i><font color="Green">E73</font></i></a>, <a class="ref" href="group_6.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i>, <a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="group_6.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">x'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font>
<b>by </b><i/>
;<br/>
<a NAME="E9:103_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>;<br/>
<a NAME="E10:103_1"/>
( <font color="Maroon">x''</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> &amp; <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">x''</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E11:103_1"/><b>then </b>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T13" target="_self">Th13</a>, <a class="ref" href="group_6.html#T14" target="_self">Th14</a>, <a class="ref" href="group_6.html#T15" target="_self">Th15</a>, </i>;<br/>
<b>hence </b><a NAME="E12:103_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </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>
<b>then reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><a NAME="E2:103_2"/><i><font color="Green">E112</font></i>: 
( <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E3:103_2"/>
( <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T11" target="_self">Th11</a>, <a class="ref" href="group_6.html#T12" target="_self">Th12</a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
;<br/><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E11:103"/><i><font color="Green">E113</font></i>: 
 <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font> <a href="group_2.html#R1">=</a> <font color="Maroon">B</font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_2.html#D6">GROUP_2:def 6</a><br/>

<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<i><font color="Green">E114</font></i>: <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font>
<b>by </b><i/>
;<br/>
<b>thus </b><a NAME="E3:103_3"/>
( <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font> implies <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">N</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:103_3_2"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font>
 ;<br/>

<b>then </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E3:103_3_2"/><b>then </b>
( <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#T13" target="_self">Th13</a>, <a class="ref" href="group_2.html#T136">GROUP_2:136</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:103_3_2"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>


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

<b>assume </b><a NAME="E4:103_3"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E5:103_3"/><b>then </b>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<b>then </b><font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> = 
 <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</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"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E7:103_3"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="group_6.html#T13" target="_self">Th13</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:103"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:103_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:103_4"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span><b> such that </b><br/><a NAME="E4:103_4"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:103_4"/>
( <font color="Maroon">B</font> <a href="group_4.html#K7">*</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_4.html#K7">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="group_5.html#T8">GROUP_5:8</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">a2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:103_4"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E8:103_4"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">a1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="ref" href="group_5.html#T5">GROUP_5:5</a></i>;<br/>
<a NAME="E9:103_4"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#T15" target="_self">Th15</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a>, <a class="ref" href="group_6.html#T13" target="_self">Th13</a>, <a class="ref" href="group_6.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_6.html#T15" target="_self">Th15</a>, <a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="group_6.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E11:103_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="group_6.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:103"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E14:103"/><b>then </b>
<font color="Maroon">g</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<b>by </b><i><a class="txt" href="group_6.html#E4:103"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E15:103"/><b>then </b>
 <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E16:103"/>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">N</font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>,<font color="Maroon">B</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">N</font></span>)</span> <a href="group_6.html#R2" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="group_6.html#T12" target="_self">Th12</a>, </i>;<br/>


</div>
