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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 ;<br/>

<a NAME="E2:13"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">n</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="c1">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> = <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:13_1"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E3:13_1"/><i><font color="Green" title="E17">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being   <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>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E4:13_1"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="setwop_2.html#K3" title="SETWOP_2:func.3">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:13_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:13_1"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E19"><i><font color="Green" title="E6">Lm2</font></i></a></i>;<br/>
<a NAME="E6:13_1"/><i><font color="Green" title="E20">A6</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <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>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K3" title="SETWOP_2:func.3">dom</a> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E4:13_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c3">F</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">y</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/>
<b>reconsider </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">x</font> as    <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>  <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:13_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="gr_cy_1.html#E5:13_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="gr_cy_1.html#E4:13_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">n</font>
;<br/>

<b>thus </b><a NAME="E6:13_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:13_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_1.html#E4:13_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:13_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="gr_cy_1.html#E4:13_1_1"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:13_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<a NAME="E2:13_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E6:13_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">F'</font> = <font color="Maroon" title="c3">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:13_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">c</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:13_1"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:13"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E11:13_1"/><i><font color="Green" title="E22">A10</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E10:13_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="card_1.html#D11" title="CARD_1:def.11">CARD_1:def 11</a></i>;<br/>
<a NAME="E12:13_1"/><i><font color="Green" title="E23">A11</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></i>;<br/>
<a NAME="E13:13_1"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E14:13_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E11:13_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="gr_cy_1.html#E12:13_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="card_1.html#T73" title="CARD_1:th.73">CARD_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:13_1"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E17:13_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E18:13_1"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font>
 <b>and </b><br/><a NAME="E19:13_1"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c4">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T80" title="FINSEQ_4:th.80">FINSEQ_4:80</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c7">y</font> as    <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>  <b>by </b><i><a class="txt" href="gr_cy_1.html#E16:13_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="gr_cy_1.html#E17:13_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="gr_cy_1.html#E5:13_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_3_1"/>
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">p</font> or <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">n</font> )
 </span><b>by </b><i><a class="txt" href="gr_cy_1.html#E18:13_1"><i><font color="Green" title="E26">A14</font></i></a>, <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="13_1_3_1_1"><b>suppose </b></a><a NAME="E1:13_1_3_1_1"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">p</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c9">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c8">p</font> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">e</font> = 0 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><b>take </b><font color="Maroon" title="c12">t</font> = <font color="Maroon" title="c10">t</font>;<br/><a NAME="E4:13_1_3_1_1"/><i><font color="Green" title="E29">A17</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:13_1_3_1_1"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/><a NAME="E5:13_1_3_1_1"/>
<font color="Maroon" title="c4">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:13_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_1.html#E16:13_1"><i><font color="Green" title="E24">A12</font></i></a></i>;<br/><a NAME="E6:13_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:13_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_1.html#E17:13_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="gr_cy_1.html#E19:13_1"><i><font color="Green" title="E27">A15</font></i></a></i>;<br/><a NAME="E7:13_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c8">p</font></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/><a NAME="E8:13_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c8">p</font></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/><a NAME="E9:13_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">t</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="c1">G</font>
 <b>by </b><i><a class="ref" href="group_1.html#T59" title="GROUP_1:th.59">GROUP_1:59</a></i>;<br/><b>hence </b><a NAME="E10:13_1_3_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">n</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="c1">G</font> )
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:13_1_3_1_1"><i><font color="Green" title="E29">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_3_1_2"><b>suppose </b></a><a NAME="E1:13_1_3_1_2"/><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c8">p</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c9">n</font> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>take </b><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c10">t</font>;<br/><b>reconsider </b><font color="Maroon" title="c12">e</font> = 0 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><a NAME="E4:13_1_3_1_2"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:13_1_3_1_2"><i><font color="Green" title="E28">A18</font></i></a></i>;<br/><a NAME="E5:13_1_3_1_2"/>
<font color="Maroon" title="c4">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:13_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_1.html#E16:13_1"><i><font color="Green" title="E24">A12</font></i></a></i>;<br/><a NAME="E6:13_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:13_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="gr_cy_1.html#E17:13_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="gr_cy_1.html#E19:13_1"><i><font color="Green" title="E27">A15</font></i></a></i>;<br/><a NAME="E7:13_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c9">n</font></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/><a NAME="E8:13_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c9">n</font></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/><a NAME="E9:13_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c11">t</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="c1">G</font>
 <b>by </b><i><a class="ref" href="group_1.html#T59" title="GROUP_1:th.59">GROUP_1:59</a></i>;<br/><b>hence </b><a NAME="E10:13_1_3_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">n</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="c1">G</font> )
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:13_1_3_1_2"><i><font color="Green" title="E29">A19</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="E22:13_1"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">n</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="c1">G</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13"/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D11" title="GROUP_1:def.11">GROUP_1:def 11</a></i>;<br/>


</div>
