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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> iff <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> iff <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y9</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="kw">thus </span><a NAME="E2:11"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:11_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E2:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
;<br/>
<a NAME="E3:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T7" onmouseover="rs('lattice3/T7')" onmouseout="rh()">LATTICE3:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:11_1"/>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T2" onmouseover="rs('lattice3/T2')" onmouseout="rh()">LATTICE3:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:11"/>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E4:11"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T2" onmouseover="rs('lattice3/T2')" onmouseout="rh()">LATTICE3:2</a></span>;<br/>
<a NAME="E5:11"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T7" onmouseover="rs('lattice3/T7')" onmouseout="rh()">LATTICE3:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:11"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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