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<b>let </b><font color="Maroon">it1</font> be  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a>, <font color="Maroon">it2</font> be  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green">E110</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>  ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; ( <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y</font> implies <font color="Olive">x'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y'</font> ) &amp; ( <font color="Olive">x'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y'</font> implies <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y</font> ) ) ) )
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green">E111</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font>  ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; ( <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y</font> implies <font color="Olive">x'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y'</font> ) &amp; ( <font color="Olive">x'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y'</font> implies <font color="Olive">x</font> <a href="lattices.html#R3">[=</a> <font color="Olive">y</font> ) ) ) )
 ;<br/>

<a NAME="E3:31_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">it1</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">it1</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it1</font></span>)</span> #)
 
;<br/>
<a NAME="E4:31_1_2"/><i><font color="Green">E113</font></i>: 
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">it2</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">it2</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it2</font></span>)</span> #)
 
;<br/>
<a NAME="E5:31_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>, <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font> : <font color="Olive">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D8">FILTER_1:def 8</a></i>;<br/>
<a NAME="E6:31_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font>, <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> : <font color="Olive">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D8">FILTER_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it1</font>
 ;<br/><b>then consider </b><font color="Maroon">p1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>, <font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font><b> such that </b><br/><a NAME="E3:31_1_2_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font>, <font color="Maroon">q1</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font> ;<br/><b>consider </b><font color="Maroon">pl1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">ql1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:31_1_2_1_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pl1</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ql1</font> &amp; ( <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> implies <font color="Maroon">pl1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql1</font> ) &amp; ( <font color="Maroon">pl1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql1</font> implies <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p1</font>, <font color="Maroon">q2</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">pl2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">ql2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:31_1_2_1_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pl2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ql2</font> &amp; ( <font color="Maroon">p2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> implies <font color="Maroon">pl2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql2</font> ) &amp; ( <font color="Maroon">pl2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql2</font> implies <font color="Maroon">p2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E10:31_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it2</font>
 <b>by </b><i>, , , </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it2</font>
 ;<br/><b>then consider </b><font color="Maroon">p1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font>, <font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font><b> such that </b><br/><a NAME="E4:31_1_2_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font>, <font color="Maroon">q1</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> ;<br/><b>consider </b><font color="Maroon">pl1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">ql1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:31_1_2_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pl1</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ql1</font> &amp; ( <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> implies <font color="Maroon">pl1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql1</font> ) &amp; ( <font color="Maroon">pl1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql1</font> implies <font color="Maroon">p1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p1</font>, <font color="Maroon">q2</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">pl2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">ql2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:31_1_2_1"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pl2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ql2</font> &amp; ( <font color="Maroon">p2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> implies <font color="Maroon">pl2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql2</font> ) &amp; ( <font color="Maroon">pl2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ql2</font> implies <font color="Maroon">p2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E11:31_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it1</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E8:31_1_2"/><b>then </b>
 <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">it2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E9:31_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i>, , , , <a class="ref" href="lattice3.html#T6">LATTICE3:6</a></i>;<br/>


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