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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E2:8"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:8"/>
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E8</font></i>: 
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:8_1"/><b>then </b><i><font color="Green">E9</font></i>: 
( ( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="orders_2.html#T38">ORDERS_2:38</a></i>;<br/>
<a NAME="E3:8_1"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="lattice7.html#E2:8_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_1_1"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lattice7.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="lattice7.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_1_2"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lattice7.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="lattice7.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice7.html#E1:8_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
(  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>


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