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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> $1];<br/>
<a NAME="E1:38_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="nat_d.html#E1:38"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E24">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">i</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> ;<br/><b>assume </b><a NAME="E1:38_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font>]
 ;<br/><a NAME="E2:38_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>hence </b><a NAME="E3:38_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="nat_d.html#E2:38"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="nat_d.html#E3:38"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="nat_d.html#E1:38_1_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:38_1"/><i><font color="Green" title="E25">A7</font></i>: 
for <font color="Olive" title="b1">i</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>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="nat_d.html#E1:38_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="nat_d.html#E2:38_1"><i><font color="Green" title="E24">A5</font></i></a>);<br/></i>
<a NAME="E4:38_1"/><b>then </b><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>]
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><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"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1"><b>suppose </b></a><a NAME="E1:38_1_2_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:38_1_2_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></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>() <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="nat_d.html#E2:38"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></i>;<br/><b>hence </b><a NAME="E3:38_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="nat_d.html#E3:38_1"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2"><b>suppose </b></a><a NAME="E1:38_1_2_2"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:38_1_2_2"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon">F<sub>2</sub></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>() <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <b>by </b><i><a class="txt" href="nat_d.html#E2:38"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></i>;<br/><a NAME="E3:38_1_2_2"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="nat_d.html#E2:38"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E4:38_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="nat_d.html#E3:38"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="nat_d.html#E4:38_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="nat_d.html#E1:38_1_2_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="nat_d.html#E2:38_1_2_2"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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