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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <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>  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"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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>  &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="nat_1.html#NM1" title="NAT_1:NM.1">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">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>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] implies  ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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>] ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c1">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> 
 <span class="kw">and </span><br/><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 ex <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>  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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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>] )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">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> <span class="kw"> such that </span><br/><a NAME="E4:21_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E6:21_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c4">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E2:21_1_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <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="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">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</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> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">m</font>) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c4">m</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/><span class="kw">hence </span><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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E2:21_1_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <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>)</span><br/><span class="kw">take </span><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="NAT_1:func.2">+</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>)</span><br/><span class="kw">thus </span><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#K2" title="NAT_1:func.2">+</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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:21_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:21_1"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S7" title="NAT_1:sch.7">NAT_1:sch 7</a>(<a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>($1);<br/>
<a NAME="E4:21_1"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:21_1"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
( <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="nat_1.html#NM1" title="NAT_1:NM.1">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> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E4:21_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:21_1"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">n</font> = <font color="Maroon" title="c2">n</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) );<br/>
<a NAME="E10:21_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">n</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E8:21_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="nat_d.html#T6" target="_self" title="NAT_D:th.6">Th6</a></span>;<br/>
<a NAME="E11:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A15</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<a NAME="E12:21_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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>  &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="nat_1.html#NM1" title="NAT_1:NM.1">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">l</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">l</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">S<sub>3</sub></font>[<font color="Maroon" title="c4">l</font>] implies  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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>] ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_2"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:21_1_2"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<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>)
 <span class="kw">and </span><br/><a NAME="E3:21_1_2"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>] )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:21_1_2"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">m</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">m</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">m</font>] )</span><br/>

<a NAME="E7:21_1_2"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E8:21_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#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<span class="kw">hence </span><a NAME="E9:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E7:21_1_2"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">m</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E32">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_2_1"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:21_1_2_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a></span>;<br/><a NAME="E3:21_1_2_1_1"/><span class="kw">then </span>
<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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_2_1_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:21_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  );<br/><a NAME="E5:21_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a></span>;<br/><a NAME="E6:21_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
 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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c7">m</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
( <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">and </span><br/><a NAME="E2:21_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:21_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:21_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:21_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></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></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c8">l</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/><span class="kw">hence </span><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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="txt" href="#E2:21_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E4:21_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:21_1_2_1_1"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="txt" href="#E6:21_1_2_1_1"><span class="lab"><font color="Green" title="E37">A27</font></span></a>);<br/></span><a NAME="E8:21_1_2_1_1"/><span class="kw">then </span>
<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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_2_1_1"><span class="lab"><font color="Green" title="E35">A25</font></span></a></span>;<br/><span class="kw">hence </span><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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T6" target="_self" title="NAT_D:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_2_2"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><a NAME="E2:21_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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/><a NAME="E3:21_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="nat_1.html#K3" title="NAT_1: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1: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#K2" title="NAT_1:func.2">+</a> 2</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_2_2"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>;<br/><a NAME="E4:21_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="nat_1.html#K3" title="NAT_1: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_2"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E5:21_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="nat_d.html#T9" target="_self" title="NAT_D:th.9">Th9</a></span>;<br/><span class="kw">hence </span><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>(<font color="Maroon" title="c6">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_2"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E5:21_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E3:21_1_2_2"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="ref" href="nat_d.html#T8" target="_self" title="NAT_D:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:21_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_1_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">thus </span><a NAME="E13:21_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_2"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E5:21_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S7" title="NAT_1:sch.7">NAT_1:sch 7</a>(<a class="txt" href="#E11:21_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E12:21_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>);<br/></span><span class="kw">hence </span><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>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="Olive" title="b1">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="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Olive" title="b1">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>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>() implies <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>) )</span><br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) );<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:21_1_3"/><span class="lab"><font color="Green" title="E27">A34</font></span>: 
<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>()
 <span class="kw">and </span><br/><a NAME="E4:21_1_3"/><span class="lab"><font color="Green" title="E28">A35</font></span>: 
<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>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><a NAME="E5:21_1_3"/><span class="lab"><font color="Green" title="E29">A36</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:21_1_3"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="txt" href="#E4:21_1_3"><span class="lab"><font color="Green" title="E28">A35</font></span></a></span>;<br/><a NAME="E6:21_1_3"/><span class="lab"><font color="Green" title="E30">A37</font></span>: 
 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>   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#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_3_1"/><span class="lab"><font color="Green" title="E31">A38</font></span>: 
<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>)
 <span class="kw">and </span><br/><a NAME="E2:21_1_3_1"/><span class="lab"><font color="Green" title="E32">A39</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">thus </span><a NAME="E3:21_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_3_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span><br/>

<a NAME="E4:21_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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_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#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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_3_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:21_1_3_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="nat_d.html#T11" target="_self" title="NAT_D:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:21_1_3"/>
 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">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:21_1_3"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E6:21_1_3"><span class="lab"><font color="Green" title="E30">A37</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E8:21_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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:21_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#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">thus </span><a NAME="E15:21_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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E8:21_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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