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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>2</sub></font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a<sub>2</sub></font> ) );<br/>
<a NAME="E1:75_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>proof </b></a><div class="add">

<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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>take </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
;<br/><b>thus </b><a NAME="E2:75_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:75_1_1_1_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<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/><div class="add"><b>take </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E2:75_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:75_1_1_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:75_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="lattice5.html#E1:75_1_1"><i><font color="Green">E56</font></i></a>);<br/></i>
<a NAME="E4:75_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:75_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="lattice5.html#NM3">BiFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:75_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="lattice5.html#V2" target="_self">zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D7" target="_self">LATTICE5:def 7</a><br/>

<b>thus </b><a NAME="E1:75_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:75_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="lattice5.html#V1" target="_self">symmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D6" target="_self">LATTICE5:def 6</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1_1_3_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_1_1_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_3_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_3_1_2"><i><font color="Green">E61</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:75_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="lattice5.html#V3" target="_self">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" target="_self">LATTICE5:def 8</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1_1_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_4_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_1_1_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:75_1_1_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_4_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:75_1_1_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_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="E1:75_1_1_4_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_4_1_2_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_1_1_4_1_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/><a NAME="E3:75_1_1_4_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:75_1_1_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>;<br/><a NAME="E5:75_1_1_4_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2_1_1_1"><i><font color="Green">E62</font></i></a></i>;<br/><b>hence </b><a NAME="E6:75_1_1_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_4_1_2_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_1_1_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_4_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_2_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="E1:75_1_1_4_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_4_1_2_1_1_2_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_1_1_4_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/><a NAME="E3:75_1_1_4_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:75_1_1_4_1_2_1_1_2_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>;<br/><b>hence </b><a NAME="E4:75_1_1_4_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2_1_1_2_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_4_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_4_1_2_1_1_2_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><a NAME="E3:75_1_1_4_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><b>hence </b><a NAME="E4:75_1_1_4_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2_1_1_2_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:75_1_1_4_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2_1_1_2"><i><font color="Green">E62</font></i></a></i>;<br/><b>hence </b><a NAME="E4:75_1_1_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice5.html#E1:75_1_1_4_1_2"><i><font color="Green">E61</font></i></a></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>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="lattice5.html#NM4">distance_function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E6:75_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice5.html#E7:75_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E10:75_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="lattice5.html#E4:75_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>


</div>
