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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] );<br/>
<div><i><font color="Green" title="E4">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:3_1_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:3_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/><b>reconsider </b><font color="Maroon" title="c2">m</font> = <font color="Maroon" title="c1">k</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="uniroots.html#E1:3_1_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/><a NAME="E2:3_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E3:3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E2:3_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="uniroots.html#E1:3_1_1_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/></div><b>end;</b></div><a NAME="E5:3_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E6:3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:3_1"/><i><font color="Green" title="E5">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9" title="NAT_1:sch.9">NAT_1:sch 9</a>(<a class="txt" href="uniroots.html#E1:3_1"><i><font color="Green" title="E4">A2</font></i></a>);<br/></i>
<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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E3:3_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E4:3_1"/><b>then </b>
 ex <font color="Olive" title="b1">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] )
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:3_1"><i><font color="Green" title="E5">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/>


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