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<b>let </b><font color="Maroon">LR1</font> be  <a href="orders_2.html#V1">strict</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">LR2</font> be  <a href="orders_2.html#V1">strict</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_2"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E2:46_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR1</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> iff  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">x</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">y</font>,<font color="Maroon">L</font> )
 <b>and </b><br/><a NAME="E3:46_1_2"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR2</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR2</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> iff  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">x</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">y</font>,<font color="Maroon">L</font> )
 ;<br/>

<b>set </b><font color="Maroon">cLR1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR1</font>;<br/>
<b>set </b><font color="Maroon">cLR2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:46_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR1</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_1_2_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<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">LR1</font> <b>by </b><i/>;<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">LR2</font> <b>by </b><i>, </i>;<br/><a NAME="E6:46_1_2_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E7:46_1_2_1_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">y</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E8:46_1_2_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:46_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR2</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:46_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR2</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:46_1_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<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">LR2</font> <b>by </b><i/>;<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">LR1</font> <b>by </b><i>, </i>;<br/><a NAME="E7:46_1_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E8:46_1_2_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">y</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E9:46_1_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:46_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR1</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:46_1_2"/>
<font color="Maroon">LR1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LR2</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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