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

<b>let </b><font color="Maroon" title="c1">Gc</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E15">A1</font></i>: 
not <font color="Maroon" title="c1">Gc</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green" title="E16">A2</font></i>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  ex <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">i</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T41" title="GR_CY_1:th.41">GR_CY_1:41</a></i>;<br/>
<a NAME="E4:17"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ,<font color="Maroon" title="c1">Gc</font> st <font color="Olive" title="b1">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">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" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> = <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font><b> such that </b><br/><a NAME="E2:17_1"/><i><font color="Green" title="E17">A3</font></i>: 
for <font color="Olive" title="b1">j1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  holds  <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j1</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:17_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:17"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="gr_cy_2.html#T11" target="_self" title="GR_CY_2:th.11">Th11</a></i>;<br/>
<a NAME="E4:17_1"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:17_1"/><i><font color="Green" title="E20">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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" title="E21">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:17_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:17_1_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:17"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">i'</font> = <font color="Maroon" title="c6">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  <b>by </b><i><a class="ref" href="gr_cy_1.html#D4" title="GR_CY_1:def.4">GR_CY_1:def 4</a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E6:17_1_1_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_1_1_1_1"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c7">i'</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8" title="GR_CY_1:def.8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E2:17_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i'</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:17_1_1_1"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_1_1_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:17_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:17_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:17_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:17_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c3">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:17_1_2"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E2:17_1_2"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E3:17_1_2"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <b>and </b><br/><a NAME="E4:17_1_2"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:17_1_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E26">A14</font></i>: <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c6">x'</font></span>)</span> = 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y'</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:17_1_2"><i><font color="Green" title="E24">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c7">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a></i>
;<br/>
<a NAME="E8:17_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">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" title="c8">i</font> =  <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c6">x'</font>;<br/>
<a NAME="E1:17_1_2_2"/>
 <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8" title="GR_CY_1:def.8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E2:17_1_2_2"/><b>then </b>
 <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c6">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c7">y'</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E4:17_1_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="gr_cy_1.html#D8" title="GR_CY_1:def.8">GR_CY_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_2_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i1</font> &amp; <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">i1</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E7:17_1_2"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:17_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:17"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:17_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="gr_cy_2.html#T20" target="_self" title="GR_CY_2:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_1"/>
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">j1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  holds  <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Olive" title="b2">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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" title="c4">j</font>, <font color="Maroon" title="c5">j1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ;<br/>



<a NAME="E1:17_1_3"/><i><font color="Green" title="E22">A15</font></i>: 
 <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8" title="GR_CY_1:def.8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E2:17_1_3"/>
 <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j1</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D8" title="GR_CY_1:def.8">GR_CY_1:def 8</a></i>;<br/>
<a NAME="E3:17_1_3"/><b>then </b>
 <a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c5">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:17_1_3"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="gr_cy_1.html#D8" title="GR_CY_1:def.8">GR_CY_1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">j1</font></span>)</span> = 
<font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c5">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c5">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> <font color="Maroon" title="c5">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:17_1"><i><font color="Green" title="E17">A3</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:17_1_3"/>
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">h</font> = <font color="Maroon" title="c3">h</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ,<font color="Maroon" title="c1">Gc</font> <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E9:17_1"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E5:17_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="group_6.html#D13" title="GROUP_6:def.13">GROUP_6:def 13</a></i>;<br/>
<a NAME="E10:17_1"/><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E6:17_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="group_6.html#D12" title="GROUP_6:def.12">GROUP_6:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">h</font>
;<br/>

<b>thus </b><a NAME="E11:17_1"/>
<font color="Maroon" title="c4">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E9:17_1"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="gr_cy_2.html#E10:17_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">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#K2" title="GR_CY_1:func.2">INT.Group</a> ,<font color="Maroon" title="c1">Gc</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a></i>;<br/>


</div>
