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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E40">A2</font></i>: 
 <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> is  not  <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/>

<a NAME="E2:49"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>  holds  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>

<a NAME="E1:49_1"/><i><font color="Green" title="E42">A5</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T28" title="GR_CY_1:th.28">GR_CY_1:28</a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E2:49_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:49"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T43" title="GR_CY_1:th.43">GR_CY_1:43</a></i>;<br/>
<b>hence </b><a NAME="E3:49_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E1:49_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">ZP</font> =  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>] <b>means </b> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> $3;<br/>
<a NAME="E4:49"/><i><font color="Green" title="E42">A6</font></i>: 
for <font color="Olive" title="b1">n</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> <br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>  ex <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>



<b>set </b><font color="Maroon" title="c6">n</font> =  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font>;<br/>
<a NAME="E1:49_2"/><i><font color="Green" title="E43">A7</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E2:49"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E2:49_2"/><i><font color="Green" title="E44">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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 <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E3:49_2"/><i><font color="Green" title="E45">A9</font></i>: 
 <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:49_2"><i><font color="Green" title="E43">A7</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/>
<a NAME="E4:49_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E2:49_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="int_7.html#E3:49_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E5:49_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T105" title="XBOOLE_1:th.105">XBOOLE_1:105</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:49_2"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c7">z</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 <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:49_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E9:49_2"/><b>then </b>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>;<br/>
<a NAME="E10:49_2"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
1 <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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="int_7.html#T21" target="_self" title="INT_7:th.21">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="txt" href="int_7.html#E7:49_2"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">m</font> =  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">d</font> = <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E13:49_2"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="int_7.html#E7:49_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="int_7.html#E10:49_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">H</font> =  <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="c8">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> ;<br/>
<a NAME="E15:49_2"/><i><font color="Green" title="E49">A13</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c11">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E16:49_2"/><i><font color="Green" title="E50">A14</font></i>: 
 <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c11">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/>
<a NAME="E17:49_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E13:49_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="int_7.html#T21" target="_self" title="INT_7:th.21">Th21</a></i>;<br/>
<a NAME="E18:49_2"/><b>then </b><i><font color="Green" title="E51">A15</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E15:49_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="int_7.html#E16:49_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="group_1.html#T85" title="GROUP_1:th.85">GROUP_1:85</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1"/>
( <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1"/><i><font color="Green" title="E52">A16</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c5">x</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">z</font>;<br/><a NAME="E2:49_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/><a NAME="E3:49_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/><a NAME="E4:49_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E18:49_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E5:49_2_1_1_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="int_7.html#E1:49_2_1_1_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="int_7.html#T25" target="_self" title="INT_7:th.25">Th25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_2"/>
<font color="Maroon" title="c10">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/><a NAME="E3:49_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">A17</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/><a NAME="E4:49_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/><a NAME="E5:49_2_1_1_2"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/><b>set </b><font color="Maroon" title="c12">l</font> = <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c13">m0</font>, <font color="Maroon" title="c14">n0</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="E7:49_2_1_1_2"/><i><font color="Green" title="E54">A19</font></i>: 
( <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n0</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m0</font> &amp; <font color="Maroon" title="c14">n0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c13">m0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c14">n0</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c13">m0</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font> &amp; <font color="Maroon" title="c14">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">m0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_7.html#E3:49_2_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="int_7.html#E5:49_2_1_1_2"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="int_7.html#T17" target="_self" title="INT_7:th.17">Th17</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E9:49_2_1_1_2"/><i><font color="Green" title="E55">A20</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/><a NAME="E10:49_2_1_1_2"/><i><font color="Green" title="E56">A21</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c14">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="int_7.html#E9:49_2_1_1_2"><i><font color="Green" title="E55">A20</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">d1</font> = <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c14">n0</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="int_7.html#E10:49_2_1_1_2"><i><font color="Green" title="E56">A21</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">u</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E13:49_2_1_1_2"/><i><font color="Green" title="E57">A22</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">u</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/><a NAME="E14:49_2_1_1_2"/><i><font color="Green" title="E58">A23</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">m0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="int_7.html#E13:49_2_1_1_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">d2</font> = <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">m0</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="int_7.html#E14:49_2_1_1_2"><i><font color="Green" title="E58">A23</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>take </b><font color="Maroon" title="c19">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c16">d1</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="c8">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c18">d2</font></span>)</span>;<br/><a NAME="E16:49_2_1_1_2"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c14">n0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">n0</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E17:49_2_1_1_2"/><b>then </b><i><font color="Green" title="E59">A24</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c16">d1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n0</font>
 <b>by </b><i><a class="ref" href="int_7.html#T30" target="_self" title="INT_7:th.30">Th30</a></i>;<br/><a NAME="E18:49_2_1_1_2"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">m0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">m0</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E19:49_2_1_1_2"/><b>then </b>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c18">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m0</font>
 <b>by </b><i><a class="ref" href="int_7.html#T30" target="_self" title="INT_7:th.30">Th30</a></i>;<br/><a NAME="E20:49_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A25</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m0</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c14">n0</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="int_7.html#E17:49_2_1_1_2"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="int_7.html#T25" target="_self" title="INT_7:th.25">Th25</a></i>;<br/><a NAME="E21:49_2_1_1_2"/><i><font color="Green" title="E61">A26</font></i>: 
not  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2_1_1_2_1"/><i><font color="Green" title="E62">A27</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c20">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:49_2_1_1_2_1"/><i><font color="Green" title="E63">A28</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="int_7.html#E1:49_2_1_1_2_1"><i><font color="Green" title="E62">A27</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<i><font color="Green" title="E64">A29</font></i>: <font color="Maroon" title="c8">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="txt" href="int_7.html#E3:49_2_1_1_2_1"><i><font color="Green" title="E63">A28</font></i></a>, <a class="ref" href="group_1.html#T68" title="GROUP_1:th.68">GROUP_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="ref" href="group_1.html#T82" title="GROUP_1:th.82">GROUP_1:82</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="int_7.html#T21" target="_self" title="INT_7:th.21">Th21</a></i>
;<br/>
<a NAME="E5:49_2_1_1_2_1"/>
<font color="Maroon" title="c8">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E3:49_2_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="int_7.html#E4:49_2_1_1_2_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="ref" href="int_7.html#T29" target="_self" title="INT_7:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E6:49_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:49_2_1_1_2"/><i><font color="Green" title="E62">A30</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E21:49_2_1_1_2"><i><font color="Green" title="E61">A26</font></i></a>, <a class="ref" href="int_2.html#T25" title="INT_2:th.25">INT_2:25</a></i>;<br/><a NAME="E23:49_2_1_1_2"/><i><font color="Green" title="E63">A31</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T25" title="INT_2:th.25">INT_2:25</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">j</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E25:49_2_1_1_2"/><i><font color="Green" title="E64">A32</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="int_7.html#E23:49_2_1_1_2"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/><a NAME="E26:49_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="int_2.html#K2" title="INT_2:func.2">lcm</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="int_7.html#E3:49_2_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="int_7.html#E25:49_2_1_1_2"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><a NAME="E27:49_2_1_1_2"/><b>then </b><i><font color="Green" title="E65">A33</font></i>: 
<font color="Maroon" title="c20">j</font> is    <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="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E28:49_2_1_1_2"/>
<font color="Maroon" title="c20">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_7.html#E3:49_2_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="int_7.html#E5:49_2_1_1_2"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="int_7.html#E25:49_2_1_1_2"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="int_2.html#T4" title="INT_2:th.4">INT_2:4</a></i>;<br/><a NAME="E29:49_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="int_7.html#E27:49_2_1_1_2"><i><font color="Green" title="E65">A33</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E30:49_2_1_1_2"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c19">y</font>
 <b>by </b><i><a class="txt" href="int_7.html#E7:49_2_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="int_7.html#E20:49_2_1_1_2"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="int_7.html#E22:49_2_1_1_2"><i><font color="Green" title="E62">A30</font></i></a>, <a class="txt" href="int_7.html#E25:49_2_1_1_2"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:49_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span><b> such that </b><br/><a NAME="E6:49"/><i><font color="Green" title="E43">A34</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="int_7.html#E4:49"><i><font color="Green" title="E42">A6</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:49"/><i><font color="Green" title="E44">A35</font></i>: 
for <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>  holds  <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</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>
<div><i><font color="Green" title="E45">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">k</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/><a NAME="E1:49_3"/><i><font color="Green" title="E46">A37</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E8:49"><i><font color="Green" title="E44">A35</font></i></a></i>;<br/><a NAME="E2:49_3"/>
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E8:49"><i><font color="Green" title="E44">A35</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_3"/>
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E6:49"><i><font color="Green" title="E43">A34</font></i></a>, <a class="txt" href="int_7.html#E1:49_3"><i><font color="Green" title="E46">A37</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:49"/><i><font color="Green" title="E46">A38</font></i>: 
for <font color="Olive" title="b1">k</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>   holds <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</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>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<a NAME="E1:49_4"/><i><font color="Green" title="E47">A39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="int_7.html#E9:49"><i><font color="Green" title="E45">A36</font></i></a></i>;<br/>
<div><i><font color="Green" title="E48">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">m</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>assume </b><a NAME="E1:49_4_1"/><i><font color="Green" title="E49">A41</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">m</font>]
 ;<br/><a NAME="E2:49_4_1"/>
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E9:49"><i><font color="Green" title="E45">A36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_4_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="int_7.html#E1:49_4_1"><i><font color="Green" title="E49">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:49_4"/>
for <font color="Olive" title="b1">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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="int_7.html#E1:49_4"><i><font color="Green" title="E47">A39</font></i></a>, <a class="txt" href="int_7.html#E2:49_4"><i><font color="Green" title="E48">A40</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:49_4"/>
for <font color="Olive" title="b1">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>   holds <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:49"/><i><font color="Green" title="E47">A42</font></i>: 
for <font color="Olive" title="b1">k</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 <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">m</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>assume </b><a NAME="E1:49_5"/><i><font color="Green" title="E48">A43</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font>
 ;<br/>

<a NAME="E2:49_5"/>
<font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">k</font> is    <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="int_7.html#E1:49_5"><i><font color="Green" title="E48">A43</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">mk</font> = <font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E4:49_5"/>
<font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_7.html#E1:49_5"><i><font color="Green" title="E48">A43</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">n0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:49_5"/><i><font color="Green" title="E49">A44</font></i>: 
<font color="Maroon" title="c8">mk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">n0</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:49_5"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E6:49_5"><i><font color="Green" title="E49">A44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:49_5"/>
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="int_7.html#E10:49"><i><font color="Green" title="E46">A38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:49"/>
<font color="Maroon" title="c5">g</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="49_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:49_6_1"/><i><font color="Green" title="E48">A45</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">xx1</font> = <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c9">xx2</font> = <font color="Maroon" title="c7">x2</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>  ;<br/><a NAME="E3:49_6_1"/><i><font color="Green" title="E49">A46</font></i>: 
<font color="Maroon" title="c8">xx1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">xx2</font>
 <b>by </b><i><a class="txt" href="int_7.html#E11:49"><i><font color="Green" title="E47">A42</font></i></a>, <a class="txt" href="int_7.html#E1:49_6_1"><i><font color="Green" title="E48">A45</font></i></a></i>;<br/><a NAME="E4:49_6_1"/>
<font color="Maroon" title="c9">xx2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">xx1</font>
 <b>by </b><i><a class="txt" href="int_7.html#E11:49"><i><font color="Green" title="E47">A42</font></i></a>, <a class="txt" href="int_7.html#E1:49_6_1"><i><font color="Green" title="E48">A45</font></i></a></i>;<br/><b>hence </b><a NAME="E5:49_6_1"/>
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="txt" href="int_7.html#E3:49_6_1"><i><font color="Green" title="E49">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:49_6"/>
<font color="Maroon" title="c5">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:49"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font>, <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E14:49"/><b>then </b><i><font color="Green" title="E48">A47</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<a NAME="E15:49"/><i><font color="Green" title="E49">A48</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p2">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</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="E1:49_7"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:49_7"/><i><font color="Green" title="E50">A49</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</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="int_7.html#E3:49_7"><i><font color="Green" title="E50">A49</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">gg</font> = <font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</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>  ;<br/>
<a NAME="E6:49_7"/>
<font color="Maroon" title="c5">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E8:49"><i><font color="Green" title="E44">A35</font></i></a></i>;<br/>
<a NAME="E7:49_7"/><b>then </b>
<font color="Maroon" title="c9">gg</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_7.html#E2:49"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:49_7"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p2">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_7.html#E3:49_7"><i><font color="Green" title="E50">A49</font></i></a>, <a class="ref" href="algseq_1.html#T10" title="ALGSEQ_1:th.10">ALGSEQ_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:49"/><i><font color="Green" title="E50">A50</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="int_7.html#E15:49"><i><font color="Green" title="E49">A48</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E17:49"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="int_7.html#E14:49"><i><font color="Green" title="E48">A47</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:49"/><b>then </b>
not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>
<b>hence </b><a NAME="E19:49"/>
contradiction
 <b>by </b><i><a class="txt" href="int_7.html#E16:49"><i><font color="Green" title="E50">A50</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>


</div>
