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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">n</font>);<br/>
<a NAME="E1:50_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:50_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E2:50"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:50_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">n</font>) )
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:50_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>)
 ;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:50_1_1"/><i><font color="Green">E49</font></i>: 
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">Q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> ) &amp;  ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1 &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">m</font>) ) )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E2:50"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:50_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E2:50_1_1_1"/><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:50_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m1</font>) <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:50_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T18" target="_self">Th18</a>, </i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1;<br/><b>thus </b><a NAME="E6:50_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:50_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>by </b><i><a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:50_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7" target="_self">NAT_1:sch 7</a>(<a class="txt" href="nat_1.html#E3:50"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_1.html#E2:50_1"><i><font color="Green">E35</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<a NAME="E5:50_1"/><i><font color="Green">E88</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:50_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:50_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E2:50"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>) );<br/>
<a NAME="E11:50_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a>, , </i>;<br/>
<a NAME="E12:50_1"/><b>then </b><i><font color="Green">E97</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<a NAME="E13:50_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">l</font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) )
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:50_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<a NAME="E5:50_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E6:50_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E7:50_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T23" target="_self">Th23</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="hidden.html#R1">=</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:50_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E3:50_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:50_1_2_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="hidden.html#R1">=</a> 0 );<br/><a NAME="E5:50_1_2_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_2_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:50_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50_1_2_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(0) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:50_1_2_1_1"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2_1_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E2:50_1_2_1_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_1_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E2:50_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:50_1_2_1_1_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E2:50"><i><font color="Green">E32</font></i></a>, , , </i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:50_1_2_1_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">l</font>) <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:50_1_2_1_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_2_1_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:50_1_2_1_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" target="_self">NAT_1:sch 2</a>(, );<br/></i><a NAME="E8:50_1_2_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:50_1_2_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:50_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:50_1_2_2"/><i><font color="Green">E114</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i/>;<br/><a NAME="E3:50_1_2_2"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>) <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i/>;<br/><a NAME="E4:50_1_2_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>) <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>)
 <b>by </b><i>, , </i>;<br/><a NAME="E5:50_1_2_2"/>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>) <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)</span>)</span> &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>) )
 <b>by </b><i><a class="ref" href="nat_1.html#T22" target="_self">Th22</a>, <a class="ref" href="nat_1.html#T23" target="_self">Th23</a>, </i>;<br/><b>hence </b><a NAME="E6:50_1_2_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>)
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:50_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">m</font>)
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E11:50_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i><a class="ref" href="nat_1.html#T22" target="_self">Th22</a>, <a class="ref" href="nat_1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_3"><b>now </b></a><div class="add"><a NAME="E1:50_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[0]
 <b>from </b><i><a class="ref" href="nat_1.html#S7" target="_self">NAT_1:sch 7</a>(<a class="ref" href="nat_1.html#T18" target="_self">Th18</a>, <a class="ref" href="nat_1.html#T20" target="_self">Th20</a>);<br/></i><b>hence </b><a NAME="E2:50_1_3"/>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>3</sub></font>() )
 <b>by </b><i><a class="txt" href="nat_1.html#E2:50"><i><font color="Green">E32</font></i></a></i>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>) );<br/><b>assume </b><a NAME="E3:50_1_3"/>
( <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>3</sub></font>() )
 ;<br/><a NAME="E4:50_1_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="txt" href="nat_1.html#E2:50"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:50_1_3"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_3_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>) &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) )
 ;<br/>

<b>hence </b><a NAME="E2:50_1_3_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 ;<br/>

<a NAME="E3:50_1_3_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">k</font>) <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:50_1_3_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">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">k</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E6:50_1_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(<a class="ref" href="nat_1.html#T22" target="_self">Th22</a>, <a class="ref" href="nat_1.html#T23" target="_self">Th23</a>);<br/></i><b>hence </b><a NAME="E7:50_1_3"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:50_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K10" target="_self">hcf</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E16:50_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E3:50_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>


</div>
