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

<b>let </b><font color="Maroon">Gc</font> be  <a href="group_1.html#V1">strict</a> <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">Gc</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font>  ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T41">GR_CY_1:41</a></i>;<br/>
<a NAME="E4:17"/>
 ex <font color="Olive">h</font> being  <a href="group_6.html#NM2">Homomorphism</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ,<font color="Maroon">Gc</font> st <font color="Olive">h</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> = <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font><b> such that </b><br/><a NAME="E2:17_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">j1</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a>  holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j1</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:17_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Gc</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:17_1"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="gr_cy_1.html#K3">INT.Group</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:17_1"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:17_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font> ;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:17_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a>  <b>by </b><i><a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E6:17_1_1_1_1"/>
 ex <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">i'</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E2:17_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:17_1_1_1_1_1"/>
 ex <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:17_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Gc</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:17_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><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="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:17_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E2:17_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E3:17_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E4:17_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E72</font></i>: <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">x'</font></span>)</span> = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E8:17_1_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">i</font> =  <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">x'</font>;<br/>
<a NAME="E1:17_1_2_2"/>
 <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E2:17_1_2_2"/><b>then </b>
 <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#D8">GR_CY_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_2_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i1</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <font color="Olive">i1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:17_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_1"/>
for <font color="Olive">j</font>, <font color="Olive">j1</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a>  holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="group_1.html#K10">*</a> <font color="Olive">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ;<br/><b>let </b><font color="Maroon">j1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ;<br/>



<a NAME="E1:17_1_3"/><i><font color="Green">E73</font></i>: 
 <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E2:17_1_3"/><i><font color="Green">E82</font></i>: 
 <a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8">GR_CY_1:def 8</a></i>;<br/>
 <a href="gr_cy_1.html#K6">@'</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="group_1.html#K10">*</a> <font color="Maroon">j1</font></span>)</span> = 
<font color="Maroon">j</font> <a href="group_1.html#K10">*</a> <font color="Maroon">j1</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8">GR_CY_1:def 8</a></i>
<br/>.= 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j1</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a>, <a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>
;<br/>
<b>then </b><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="group_1.html#K10">*</a> <font color="Maroon">j1</font></span>)</span> = 
<font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K6">@'</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:17_1_3"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="group_1.html#K10">*</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ,<font color="Maroon">Gc</font> <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E9:17_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">h</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<b>by </b><i>, <a class="ref" href="group_6.html#D13">GROUP_6:def 13</a></i>;<br/>
<a NAME="E10:17_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">h</font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a> 
 
<b>by </b><i>, <a class="ref" href="group_6.html#D12">GROUP_6:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E11:17_1"/>
<font color="Maroon">h</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i>, , <a class="ref" href="group_6.html#D14">GROUP_6:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17"/>
 <a href="gr_cy_1.html#K3">INT.Group</a> ,<font color="Maroon">Gc</font> <a href="group_6.html#R2">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15">GROUP_6:def 15</a></i>;<br/>


</div>
