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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">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">b<sub>1</sub></font>);<br/>
<a NAME="E1:49_1"/><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E2:49"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E2:49_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:49_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 ;<br/>
<b>take </b>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
;<br/>

<a NAME="E4:49_1_1"/><i><font color="Green">E38</font></i>: 
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 &amp; <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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="Olive">b<sub>1</sub></font>) ) )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E2:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_1.html#E3:49_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:49_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 ;<br/><a NAME="E2:49_1_1_1"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>by </b><i><a class="ref" href="nat_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:49_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font>) <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_1.html#E3:49"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E4:49_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_1.html#E3:49_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nat_1.html#E1:49_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nat_1.html#E2:49_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nat_1.html#T46" target="_self">Th46</a></i>;<br/><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1;<br/><b>thus </b><a NAME="E5:49_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:49_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)] )
 <b>by </b><i><a class="txt" href="nat_1.html#E4:49_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:49_1"/><i><font color="Green">E36</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#E1:49_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_1.html#E2:49_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>] means 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="E4:49_1"/><i><font color="Green">E37</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E3:49_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:49_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" target="_self">NAT_1:sch 5</a>(<a class="txt" href="nat_1.html#E4:49_1"><i><font color="Green">E37</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:49_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E2:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_1.html#E6:49_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) );<br/>
<a NAME="E9:49_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="nat_1.html#E6:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nat_1.html#E8:49_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E10:49_1"/><b>then </b><i><font color="Green">E40</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<a NAME="E11:49_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:49_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:49_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E4:49_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:49_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E6:49_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:49_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nat_1.html#E4:49_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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="49_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:49_1_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E6:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nat_1.html#E1:49_1_2_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:49_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49_1_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:49_1_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38" target="_self">Th38</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>1</sub></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="E5:49_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_2_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:49_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E3:49_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#E6:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nat_1.html#E1:49_1_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:49_1_2_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E2:49_1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 ;<br/><a NAME="E2:49_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E2:49_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:49_1_2_1_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_1.html#E2:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_1.html#E1:49_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_1.html#T22" target="_self">Th22</a>, <a class="ref" href="nat_1.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E5:49_1_2_1_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font>) <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_1.html#E3:49"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E6:49_1_2_1_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#E1:49_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nat_1.html#E2:49_1_2_1_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_1.html#E4:49_1_2_1_1_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="nat_1.html#T38" target="_self">Th38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_1_2_1_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#E6:49_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:49_1_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(<a class="txt" href="nat_1.html#E5:49_1_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_1.html#E6:49_1_2_1_1"><i><font color="Green">E50</font></i></a>);<br/></i><a NAME="E8:49_1_2_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_1.html#E4:49_1_2_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><b>hence </b><a NAME="E9:49_1_2_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="ref" href="nat_1.html#T53" target="_self">Th53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="nat_1.html#E8:49_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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="E2:49_1_2_2"/><i><font color="Green">E47</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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i><a class="ref" href="nat_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:49_1_2_2"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>) <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_1.html#E3:49"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:49_1_2_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="nat_1.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="nat_1.html#K1" target="_self">+</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="nat_1.html#E2:49_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E5:49_1_2_2"/>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) <a href="nat_1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) <a href="nat_1.html#K3" target="_self">div</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 2</span>)</span>) )
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nat_1.html#E3:49_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E6:49_1_2_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="nat_1.html#E4:49_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#T55" target="_self">Th55</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:49_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="txt" href="nat_1.html#E7:49_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:49_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 <b>by </b><i><a class="txt" href="nat_1.html#E1:49_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nat_1.html#E3:49_1_2"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_3"><b>now </b></a><div class="add"><a NAME="E1:49_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="txt" href="nat_1.html#E10:49_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nat_1.html#E11:49_1"><i><font color="Green">E41</font></i></a>);<br/></i><b>hence </b><a NAME="E2:49_1_3"/>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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#E2:49"><i><font color="Green">E32</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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>] means ( <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></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="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) );<br/><b>assume </b><a NAME="E3:49_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>3</sub></font>() )
 ;<br/><a NAME="E4:49_1_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="txt" href="nat_1.html#E2:49"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:49_1_3"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:49_1_3_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>) )
 ;<br/>

<b>hence </b><a NAME="E2:49_1_3_1"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 ;<br/>

<a NAME="E3:49_1_3_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>) <a href="nat_1.html#K4" target="_self">mod</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>)
 
<b>by </b><i><a class="txt" href="nat_1.html#E3:49"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:49_1_3_1"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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><a class="txt" href="nat_1.html#E1:49_1_3_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T58" target="_self">Th58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:49_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(<a class="txt" href="nat_1.html#E4:49_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nat_1.html#E5:49_1_3"><i><font color="Green">E43</font></i></a>);<br/></i><b>hence </b><a NAME="E7:49_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1" target="_self">divides</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:49_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K8" target="_self">hcf</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="nat_1.html#D5" target="_self">Def5</a></i>;<br/>

<b>thus </b><a NAME="E14:49_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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#E6:49_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nat_1.html#E8:49_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div>
