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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">Gc</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">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><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">Gc</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:16"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Gc</font><b> such that </b><br/><a NAME="E5:16"/><i><font color="Green" title="E18">A4</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="c2">Gc</font>  ex <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T42" title="GR_CY_1:th.42">GR_CY_1:42</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c2">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="c3">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1"/>
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="c2">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="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Gc</font>;<br/>

<b>consider </b><font color="Maroon" title="c5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:16_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:16"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c5">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">p1</font>
;<br/>

<b>thus </b><a NAME="E4:16_1_1"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_1_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1"/>
<font color="Maroon" title="c2">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="c3">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_2.html#T11" target="_self" title="GR_CY_2:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:16"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">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="16_2"><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 <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span>) <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="c2">Gc</font> = <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">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 <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Gc</font><b> such that </b><br/><a NAME="E2:16_2"/><i><font color="Green" title="E20">A7</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span> holds  <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</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="E3:16_2"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_1.html#D6" title="GR_CY_1:def.6">GR_CY_1:def 6</a></i>;<br/>
<a NAME="E4:16_2"/><i><font color="Green" title="E22">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">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 <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<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="E5:16_2"/><i><font color="Green" title="E23">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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="c2">Gc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1"/><i><font color="Green" title="E24">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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="c2">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="E2:16_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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="c4">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:16_2_1_1"/>
<font color="Maroon" title="c5">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="c2">Gc</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Gc</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:16_2_1_1"/><i><font color="Green" title="E25">A12</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:16"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<a NAME="E5:16_2_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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="16_2_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E1:16_2_1_1_1"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<i><font color="Green" title="E27">A14</font></i>: <font color="Maroon" title="c6">z</font> = 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_1_1"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" title="GROUP_1:th.48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42" title="GROUP_1:th.42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c9">t'</font> = <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:16_2_1_1_1"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E4:16_2_1_1_1"/><i><font color="Green" title="E28">A15</font></i>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c9">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">t'</font>
;<br/>

<b>thus </b><a NAME="E5:16_2_1_1_1"/>
( <font color="Maroon" title="c9">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">t'</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_1_1_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_1_1_1"><i><font color="Green" title="E28">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_2_1_1"/>
<font color="Maroon" title="c5">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="c4">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="E3:16_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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="c2">Gc</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_1"><i><font color="Green" title="E24">A11</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="E6:16_2"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c4">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="16_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_2_2"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h</font>
 <b>and </b><br/><a NAME="E2:16_2_2"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h</font>
 <b>and </b><br/><a NAME="E3:16_2_2"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 <b>and </b><br/><a NAME="E4:16_2_2"/><i><font color="Green" title="E28">A20</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2"><i><font color="Green" title="E25">A17</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="c8">y'</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2_2"><i><font color="Green" title="E26">A18</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="E29">A21</font></i>: <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">x'</font></span>)</span> = 
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y'</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2_2"><i><font color="Green" title="E27">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green" title="E20">A7</font></i></a></i>
;<br/>
<a NAME="E8:16_2_2"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c9">p</font> =  <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">x'</font>;<br/>
<b>set </b><font color="Maroon" title="c10">m</font> =  <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">y'</font>;<br/>
<a NAME="E1:16_2_2_2"/><i><font color="Green" title="E30">A22</font></i>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x'</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></i>;<br/>
<a NAME="E2:16_2_2_2"/><b>then </b><i><font color="Green" title="E31">A23</font></i>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">x'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E3:16_2_2_2"/><i><font color="Green" title="E32">A24</font></i>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y'</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></i>;<br/>
<a NAME="E4:16_2_2_2"/><b>then </b>
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">y'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_2_2"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:16_2_2"><i><font color="Green" title="E28">A20</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:16_2_2"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:16_2_2_2"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_2_2"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2_2_2"><i><font color="Green" title="E32">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:16_2_2"/><i><font color="Green" title="E30">A25</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font>
 <b>and </b><br/><a NAME="E11:16_2_2"/><i><font color="Green" title="E31">A26</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">m</font>
 <b>and </b><br/><a NAME="E12:16_2_2"/><i><font color="Green" title="E32">A27</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E13:16_2_2"/><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/>
<a NAME="E14:16_2_2"/><i><font color="Green" title="E34">A29</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1"/>
( <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> or <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">p</font> or <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1"><b>case </b></a><a NAME="E1:16_2_2_3_1_1_1"/><i><font color="Green" title="E35">A30</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">m</font>, <font color="Maroon" title="c12">p1</font> = <font color="Maroon" title="c9">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c13">r1</font> = <font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">p1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_1"><i><font color="Green" title="E35">A30</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_1_1_1"/>
( <font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_1"/><i><font color="Green" title="E36">A31</font></i>: 
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c14">zero</font> = 0;<br/><a NAME="E2:16_2_2_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E11:16_2_2"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>;<br/><a NAME="E3:16_2_2_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> 0
 <b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>;<br/><a NAME="E4:16_2_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E37">A32</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font>
 <b>by </b><i><a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></i>;<br/><a NAME="E5:16_2_2_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E13:16_2_2"><i><font color="Green" title="E33">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_2_3_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E38">A33</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">n1</font> = <font color="Maroon" title="c1">n</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:16_2_2_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E13:16_2_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_2_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:16_2_2_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_2_3_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_1_1_1_1"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_2_3_1_1_1_1_1_1"><i><font color="Green" title="E37">A32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:16_2_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2_2_3_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2"><b>case </b></a><a NAME="E1:16_2_2_3_1_1_2"/><i><font color="Green" title="E35">A34</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">p</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">m</font>, <font color="Maroon" title="c12">p1</font> = <font color="Maroon" title="c9">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon" title="c13">r1</font> = <font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_2"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_2_1"/>
( <font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1"/><i><font color="Green" title="E36">A35</font></i>: 
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c14">zero</font> = 0;<br/><a NAME="E2:16_2_2_3_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E11:16_2_2"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>;<br/><a NAME="E3:16_2_2_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> 0
 <b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>;<br/><a NAME="E4:16_2_2_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E37">A36</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font>
 <b>by </b><i><a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></i>;<br/><a NAME="E5:16_2_2_3_1_1_2_1_1"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E12:16_2_2"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_2_3_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E38">A37</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">n1</font> = <font color="Maroon" title="c1">n</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:16_2_2_3_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E12:16_2_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_2_3_1_1_2_1_1_1_1_2"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:16_2_2_3_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_2_3_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_2_1_1"><i><font color="Green" title="E36">A35</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_2_3_1_1_2_1_1"><i><font color="Green" title="E37">A36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_2"/>
<font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:16_2_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_3"><b>case </b></a><a NAME="E1:16_2_2_3_1_1_3"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:16_2_2"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_2_2_3"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:16_2_2"/><i><font color="Green" title="E35">A38</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E16:16_2_2"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:16"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/>
<b>hence </b><a NAME="E17:16_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E14:16_2_2"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="gr_cy_2.html#E15:16_2_2"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:16_2"/>
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 <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span> holds  <font color="Maroon" title="c4">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#K1" title="GROUP_1:func.1">*</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="c4">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#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="16_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">j1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<a NAME="E1:16_2_3"/><i><font color="Green" title="E25">A39</font></i>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j'</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c8">j1'</font> = <font color="Maroon" title="c6">j1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> = 
<font color="Maroon" title="c5">j</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">j1</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j'</font>,<font color="Maroon" title="c8">j1'</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">j1'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_3"><i><font color="Green" title="E25">A39</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="newton.html#T77" title="NEWTON:th.77">NEWTON:77</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> = 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green" title="E20">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T64" title="GROUP_1:th.64">GROUP_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T71" title="GROUP_1:th.71">GROUP_1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42" title="GROUP_1:th.42">GROUP_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">Gc</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T16" title="GROUP_1:th.16">GROUP_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" title="GROUP_1:th.48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green" title="E20">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green" title="E20">A7</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:16_2_3"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">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="E9:16_2"/><i><font color="Green" title="E25">A40</font></i>: 
<font color="Maroon" title="c5">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#E5:16_2"><i><font color="Green" title="E23">A10</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="E10:16_2"/><i><font color="Green" title="E26">A41</font></i>: 
<font color="Maroon" title="c5">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#E6:16_2"><i><font color="Green" title="E24">A16</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="c5">h</font>
;<br/>

<b>thus </b><a NAME="E11:16_2"/>
<font color="Maroon" title="c5">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#E9:16_2"><i><font color="Green" title="E25">A40</font></i></a>, <a class="txt" href="gr_cy_2.html#E10:16_2"><i><font color="Green" title="E26">A41</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="E8:16"/>
 <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">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>
