<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<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">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</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">x</font>];<br/>
<a NAME="E1:12_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</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">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:12_1_1_1_1"/>
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] )
 ;<br/>

<a NAME="E2:12_1_1_1_1"/><b>then </b>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:12_1_1"/><i><font color="Green">E39</font></i>: 
for <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="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<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="E1:12_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:12_1_1_2"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:12_1_1_2_1"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:12_1_1_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_2_1_1"/>
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_1_1_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:12_1_1_2_1_1_1_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:12_1_1_2_1_1_1_1"/><b>then </b>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12_1_1_2_1_1_1_1"/><b>then </b>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:12_1_1_2_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:12_1_1"/>
for <font color="Olive">n</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">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div>
<a NAME="E2:12_1"/>
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] ) implies for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_2"/><i><font color="Green">E50</font></i>: 
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Olive">x</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] ) &amp; not for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:12_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/>
<a NAME="E4:12_1_2"/>
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:12_1_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/>


</div>
