<?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> be    <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>3</sub></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">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> ;<br/>
<a NAME="E1:103"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">E73</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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<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">E74</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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:103"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:103"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_6.html#E4:103"><i><font color="Green">E75</font></i></a>, <a class="ref" href="group_6.html#T34" target="_self">Th34</a></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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</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> 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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">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="E1:103_1"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:103_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:103_1"/><i><font color="Green">E79</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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</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> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="group_6.html#E2:103"><i><font color="Green">E73</font></i></a>, <a class="txt" href="group_6.html#E3:103_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</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_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="group_6.html#E1:103"><i><font color="Green">E72</font></i></a>, <a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<b>reconsider </b><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>1</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<i><font color="Green">E80</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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</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> = 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="group_6.html#E3:103_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#D9" target="_self">Def9</a></i>
;<br/>
<a NAME="E9:103_1"/><b>then </b><i><font color="Green">E81</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">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</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> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E5:103"><i><font color="Green">E76</font></i></a>, <a class="ref" href="group_6.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E11:103_1"/><b>then </b>
<font color="Maroon">c<sub>8</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">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="group_6.html#E4:103_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="group_6.html#E8:103_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="group_6.html#E9:103_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="group_6.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E12:103_1"/>
<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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="group_6.html#E3:103"><i><font color="Green">E74</font></i></a>, <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">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</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>1</sub></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">E78</font></i>: 
( <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></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">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> &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#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> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <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>1</sub></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">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> = 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" target="_self">Def7</a></i>
<br/>.= 
<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="group_6.html#E6:103"><i><font color="Green">E77</font></i></a>, <a class="txt" href="group_6.html#E2:103_2"><i><font color="Green">E78</font></i></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">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="group_6.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E11:103"/><i><font color="Green">E78</font></i>: 
 <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></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">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></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<sub>11</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>1</sub></font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<i><font color="Green">E79</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#D9" target="_self">Def9</a></i>
;<br/>
<b>thus </b><a NAME="E3:103_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">c<sub>9</sub></font> implies <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></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">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then </b><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> = 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T50" target="_self">Th50</a></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T29" target="_self">Th29</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="group_6.html#E5:103"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>
<a NAME="E3:103_3_2"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E2:103_3"><i><font color="Green">E79</font></i></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">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></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">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:103_3"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></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<sub>11</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>
<br/>.= 
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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E5:103"><i><font color="Green">E76</font></i></a></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>

<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T29" target="_self">Th29</a></i>
;<br/>
<b>hence </b><a NAME="E7:103_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="group_6.html#E2:103_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="group_6.html#T50" target="_self">Th50</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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>9</sub></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">c<sub>10</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="E1:103_4"/>
<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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>10</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">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>11</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_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:103_4"/><i><font color="Green">E79</font></i>: 
( <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_2.html#K12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="group_6.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E5:103_4"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K7">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_4.html#K7">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:103_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E8:103_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></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">E82</font></i>: 
( <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> &amp; <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>1</sub></font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E8:103_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="group_6.html#E5:103"><i><font color="Green">E76</font></i></a>, <a class="txt" href="group_6.html#E4:103_4"><i><font color="Green">E79</font></i></a>, <a class="txt" href="group_6.html#E7:103_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="group_6.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="group_2.html#K3">*</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="txt" href="group_6.html#E8:103_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="group_6.html#D9" target="_self">Def9</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="group_6.html#E9:103_4"><i><font color="Green">E82</font></i></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">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="group_6.html#E3:103"><i><font color="Green">E74</font></i></a>, <a class="txt" href="group_6.html#E9:103_4"><i><font color="Green">E82</font></i></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>9</sub></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">c<sub>9</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E15:103"/><b>then </b>
 <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T67" target="_self">Th67</a></i>;<br/>
<b>hence </b><a NAME="E16:103"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#K1" target="_self">`*`</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_6.html#R2" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="group_6.html#E11:103"><i><font color="Green">E78</font></i></a>, <a class="txt" href="group_6.html#E107"><i><font color="Green">Lemma71</font></i></a></i>;<br/>


</div>
