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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>( not $1 is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:32_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="nat_2.html#D1" target="_self" title="NAT_2:def.1">Def1</a></i>;<br/>
<div><i><font color="Green" title="E12">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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="E1:32_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1"/>
not <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="nat_2.html#E1:32"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c1">k</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="nat_2.html#D1" target="_self" title="NAT_2:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="nat_2.html#E2:32"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="nat_2.html#E1:32_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:32_1"/>
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">k</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="nat_2.html#E1:32_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="nat_2.html#E2:32_1"><i><font color="Green" title="E12">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:32_1"/>
for <font color="Olive" title="b1">k</font> being non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <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">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>


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