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<b>consider </b><font color="Maroon" title="c1">n</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> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] )
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>
<a NAME="E3:3_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:3_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:3_1"/><b>then </b><i><font color="Green" title="E5">A4</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 
;<br/>
<a NAME="E5:3_1"/><i><font color="Green" title="E6">A5</font></i>: 
for <font color="Olive" title="b1">k</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<a NAME="E6:3_1"/>
for <font color="Olive" title="b1">k</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="hallmar1.html#S1" target="_self" title="HALLMAR1:sch.1">HALLMAR1:sch 1</a>(<a class="txt" href="hallmar1.html#E4:3_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="hallmar1.html#E5:3_1"><i><font color="Green" title="E6">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[1]
 <b>by </b><i><a class="txt" href="hallmar1.html#E2:3_1"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/>


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