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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E1:34_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="ami_3.html#E1:34"><i><font color="Green">E29</font></i></a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>now </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="E1:34_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><a NAME="E2:34_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>hence </b><a NAME="E3:34_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="ami_3.html#E2:34"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ami_3.html#E3:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ami_3.html#E1:34_1_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:34_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="ami_3.html#E1:34_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ami_3.html#E2:34_1"><i><font color="Green">E33</font></i></a>);<br/></i>
<a NAME="E4:34_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>]
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_2"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1"><b>suppose </b></a><a NAME="E1:34_1_2_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_2_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="ami_3.html#E2:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><b>hence </b><a NAME="E3:34_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="ami_3.html#E3:34_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_2"><b>suppose </b></a><a NAME="E1:34_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="txt" href="ami_3.html#E2:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E3:34_1_2_2"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_1.html#K4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="ami_3.html#E2:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><b>hence </b><a NAME="E4:34_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="ami_3.html#E3:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ami_3.html#E4:34_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ami_3.html#E1:34_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ami_3.html#E2:34_1_2_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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