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<div class="add">

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

<b>set </b><font color="Maroon">fm</font> =  <a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,0;<br/>
<b>set </b><font color="Maroon">fd</font> =  <a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0;<br/>
<b>set </b><font color="Maroon">g</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0;<br/>
<a NAME="E1:38"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:38"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E3:38_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> 0
<b>by </b><i/>
<br/>.= 
0

;<br/>


</div><b>end;</b></div>
<a NAME="E4:38_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:38_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="funct_2.html#K3">.</a> 1 = 
0 <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E3:38_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="real_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_1_2_1_2"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="txt" href="real_3.html#E3:38_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="real_3.html#E4:38_1"><i><font color="Green">E33</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:38_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, <a class="txt" href="real_3.html#E1:38"><i><font color="Green">E31</font></i></a>);<br/></i>
<a NAME="E6:38_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E7:38_1"/>
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
 <a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,0 <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0
 <b>by </b><i><a class="txt" href="real_3.html#E2"><i><font color="Green">Lemma29</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
