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<div class="add">

<b>let </b><font color="Maroon">R1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">R2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green">E41</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R2</font> #)
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">R1</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>

<a NAME="E3:81"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">R2</font> is <a href="lattice3.html#V2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_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">R2</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font><b> such that </b><br/><a NAME="E3:81_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font> )
 <b>and </b><br/><a NAME="E4:81_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">w'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>  st <font color="Olive">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Olive">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font> holds <br/><font color="Olive">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E6:81_1"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="txt" href="waybel11.html#E2:81"><i><font color="Green">E53</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>;<br/>

<b>reconsider </b><font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E8:81_1"/>
( <font color="Maroon">w</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E9:81_1"/><b>then </b>
( <font color="Maroon">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:81_1"/><b>then </b>
<font color="Maroon">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:81"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:81_1"/>
<font color="Maroon">w</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:81"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">R2</font> is <a href="lattice3.html#V1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><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">R2</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font><b> such that </b><br/><a NAME="E3:81_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">z'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">z'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y'</font> )
 <b>and </b><br/><a NAME="E4:81_2"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">w'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>  st <font color="Olive">w'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font> &amp; <font color="Olive">w'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y'</font> holds <br/><font color="Olive">w'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E6:81_2"/>
( <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="txt" href="waybel11.html#E3:81"><i><font color="Green">E54</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>;<br/>

<b>reconsider </b><font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E8:81_2"/>
( <font color="Maroon">w</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E9:81_2"/><b>then </b>
( <font color="Maroon">w'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">w'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:81_2"/><b>then </b>
<font color="Maroon">w'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E11:81_2"/>
<font color="Maroon">w</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:81"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">R2</font> is <a href="orders_2.html#V2">reflexive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><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">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<a NAME="E2:81_3"/>
<font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:81_3"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:81"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">R2</font> is <a href="orders_2.html#V3">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4"><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">R2</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:81_4"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E3:81_4"/><b>then </b>
( <font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:81_4"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:81_4"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:81"/>
<font color="Maroon">R2</font> is <a href="orders_2.html#V4">antisymmetric</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_5"><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">R2</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:81_5"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E3:81_5"/><b>then </b>
( <font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:81_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:81"/>
<font color="Maroon">R2</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i>, <a class="txt" href="waybel11.html#E2:81"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel11.html#E3:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel11.html#T1" target="_self">Th1</a></i>;<br/>


</div>
