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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </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 $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">n</font>);<br/>
<a NAME="E1:21_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E2:21_1"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
 ex <font color="Olive" title="b1">k</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">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="nat_d.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E3:21_1"/><i><font color="Green" title="E22">A5</font></i>: 
for <font color="Olive" title="b1">k</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">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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 <br/>( <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp;  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="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">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> <b> such that </b><br/><a NAME="E3:21_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">n</font>)
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">Q</font> = <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) 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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:21_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
( ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Maroon" title="c3">Q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> ) &amp;  ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">m</font>) ) )
 
<b>by </b><i><a class="txt" href="nat_d.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="nat_d.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="nat_d.html#E3:21_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E1:21_1_1_1"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E2:21_1_1_1"/><i><font color="Green" title="E27">A10</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">n</font>)
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c4">m</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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:21_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">m1</font>) <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_d.html#E3:21"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:21_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="nat_d.html#E3:21_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="nat_d.html#E1:21_1_1_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="nat_d.html#E2:21_1_1_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th46</a></i>;<br/><b>take </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>thus </b><a NAME="E6:21_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:21_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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] )
 <b>by </b><i><a class="txt" href="nat_d.html#E5:21_1_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_1"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7" title="NAT_1:sch.7">NAT_1:sch 7</a>(<a class="txt" href="nat_d.html#E2:21_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="nat_d.html#E3:21_1"><i><font color="Green" title="E22">A5</font></i></a>);<br/></i>
<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>0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>($1);<br/>
<a NAME="E5:21_1"/><i><font color="Green" title="E24">A12</font></i>: 
 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">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>by </b><i><a class="txt" href="nat_d.html#E4:21_1"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c1">k</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> <b> such that </b><br/><a NAME="E7:21_1"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">k</font>] &amp; ( for <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">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="nat_d.html#E5:21_1"><i><font color="Green" title="E24">A12</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:21_1"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="nat_d.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="nat_d.html#E7:21_1"><i><font color="Green" title="E25">A13</font></i></a>, <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="c3">n</font> = <font color="Maroon" title="c2">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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">n</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>($1) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) );<br/>
<a NAME="E11:21_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">n</font>]
 
<b>by </b><i><a class="txt" href="nat_d.html#E7:21_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="nat_d.html#E9:21_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="nat_d.html#T6" target="_self" title="NAT_D:th.6">Th53</a></i>;<br/>
<a NAME="E12:21_1"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
 ex <font color="Olive" title="b1">k</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">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<a NAME="E13:21_1"/><i><font color="Green" title="E28">A16</font></i>: 
for <font color="Olive" title="b1">k</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">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/> ex <font color="Olive" title="b2">m</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="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">m</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">l</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:21_1_2"/><i><font color="Green" title="E29">A17</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">l</font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:21_1_2"/><i><font color="Green" title="E30">A18</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">m</font> = <font color="Maroon" title="c5">m</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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">m</font>
;<br/>

<a NAME="E5:21_1_2"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</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#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:21_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E7:21_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font>
 <b>by </b><i><a class="txt" href="nat_d.html#E3:21_1_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="nat_d.html#E5:21_1_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>

<div><i><font color="Green" title="E32">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1"/><i><font color="Green" title="E33">A21</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1_1"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font>
 ;<br/><a NAME="E2:21_1_2_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="nat_d.html#E7:21_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="nat_d.html#E1:21_1_2_1"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/><a NAME="E3:21_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21_1_2_1_1"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:21_1_2_1_1"/><b>then </b><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies <font color="Maroon">F<sub>1</sub></font>($1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 );<br/><a NAME="E5:21_1_2_1_1"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="txt" href="nat_d.html#E7:21_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:21_1_2_1_1"/><i><font color="Green" title="E37">A26</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_2_1_1_1"/><i><font color="Green" title="E38">A27</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> implies <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>and </b><br/><a NAME="E2:21_1_2_1_1_1"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1_1_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E2:21_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="nat_d.html#E2:21_1_2_1_1_1"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:21_1_2_1_1_1_1"/><i><font color="Green" title="E41">A30</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="nat_d.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="nat_d.html#E1:21_1_2_1_1_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c8">l</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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:21_1_2_1_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c9">l</font>) <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_d.html#E3:21"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E7:21_1_2_1_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21_1_2_1_1_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="nat_d.html#E2:21_1_2_1_1_1_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="nat_d.html#E4:21_1_2_1_1_1_1"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="nat_d.html#D2" target="_self" title="NAT_D:def.2">Def2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_2_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E7:21_1"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:21_1_2_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="nat_d.html#E5:21_1_2_1_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="nat_d.html#E6:21_1_2_1_1"><i><font color="Green" title="E37">A26</font></i></a>);<br/></i><a NAME="E8:21_1_2_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E4:21_1_2_1_1"><i><font color="Green" title="E35">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E9:21_1_2_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>)
 <b>by </b><i><a class="ref" href="nat_d.html#T6" target="_self" title="NAT_D:th.6">Th53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>)
 <b>by </b><i><a class="txt" href="nat_d.html#E9:21_1"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_2"/><i><font color="Green" title="E33">A31</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:21_1_2_2"/><i><font color="Green" title="E34">A32</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 ;<br/><a NAME="E3:21_1_2_2"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>) <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_d.html#E3:21"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/><a NAME="E4:21_1_2_2"/><b>then </b><i><font color="Green" title="E35">A33</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>) <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21_1_2_2"><i><font color="Green" title="E33">A31</font></i></a>, <a class="txt" href="nat_d.html#E2:21_1_2_2"><i><font color="Green" title="E34">A32</font></i></a>, <a class="ref" href="nat_d.html#T2" target="_self" title="NAT_D:th.2">Th47</a></i>;<br/><a NAME="E5:21_1_2_2"/>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>) <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) )
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21_1_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="nat_d.html#E3:21_1_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="nat_d.html#T9" target="_self" title="NAT_D:th.9">Th56</a></i>;<br/><b>hence </b><a NAME="E6:21_1_2_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>)
 <b>by </b><i><a class="txt" href="nat_d.html#E4:21_1_2_2"><i><font color="Green" title="E35">A33</font></i></a>, <a class="ref" href="nat_d.html#T8" target="_self" title="NAT_D:th.8">Th55</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:21_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c6">m</font>)
 <b>by </b><i><a class="txt" href="nat_d.html#E8:21_1_2"><i><font color="Green" title="E32">A20</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:21_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_d.html#E1:21_1_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="nat_d.html#E3:21_1_2"><i><font color="Green" title="E30">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3"><b>now </b></a><div class="add"><a NAME="E1:21_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[0]
 <b>from </b><i><a class="ref" href="nat_1.html#S7" title="NAT_1:sch.7">NAT_1:sch 7</a>(<a class="txt" href="nat_d.html#E12:21_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="nat_d.html#E13:21_1"><i><font color="Green" title="E28">A16</font></i></a>);<br/></i><b>hence </b><a NAME="E2:21_1_3"/>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>3</sub></font>() )
 <b>by </b><i><a class="txt" href="nat_d.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>($1) &amp; <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) );<br/><b>assume </b><a NAME="E3:21_1_3"/>
( <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>3</sub></font>() )
 ;<br/><a NAME="E4:21_1_3"/><b>then </b><i><font color="Green" title="E29">A34</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="txt" href="nat_d.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><a NAME="E5:21_1_3"/><i><font color="Green" title="E30">A35</font></i>: 
for <font color="Olive" title="b1">k</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">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:21_1_3_1"/><i><font color="Green" title="E31">A36</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">k</font>) &amp; <font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 ;<br/>

<b>hence </b><a NAME="E2:21_1_3_1"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 ;<br/>

<a NAME="E3:21_1_3_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">k</font>) <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 
<b>by </b><i><a class="txt" href="nat_d.html#E3:21"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_3_1"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="nat_d.html#E1:21_1_3_1"><i><font color="Green" title="E31">A36</font></i></a>, <a class="ref" href="nat_d.html#T11" target="_self" title="NAT_D:th.11">Th58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:21_1_3"/>
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</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="nat_d.html#E4:21_1_3"><i><font color="Green" title="E29">A34</font></i></a>, <a class="txt" href="nat_d.html#E5:21_1_3"><i><font color="Green" title="E30">A35</font></i></a>);<br/></i><b>hence </b><a NAME="E7:21_1_3"/>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:21_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></i>;<br/>

<b>thus </b><a NAME="E16:21_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E7:21_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="nat_d.html#E9:21_1"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/>


</div>
