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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">n</font>];<br/>
<a NAME="E1:5_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>]
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:5_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E3:5_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1"/>
(  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> or  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_1"/>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>] )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:5_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>]
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_2"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:5_1_1_1_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E2:5_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E3:5_1_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E4:5_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1_1_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>]
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:5_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1_2_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_2_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_1_1_2_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_2_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>]
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:5_1_1_1_2_2_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:5_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">k</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E3:5_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:5_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">n</font>]
 <b>by </b><i><a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>


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