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<a NAME="E1:3_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c2">x</font> =  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</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="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E2:3_1_1_1"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Maroon" title="c3">z</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Olive" title="b1">v</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D13" onmouseover="rs('lattices/D13')" onmouseout="rh()">LATTICES:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">z9</font> = <font color="Maroon" title="c3">z</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#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<a NAME="E4:3_1_1_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1_1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="lattices.html#D16" onmouseover="rs('lattices/D16')" onmouseout="rh()">LATTICES:def 16</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E3">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">b9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">b9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Maroon" title="c4">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b9</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</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="c5">b9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b9</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">b9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><span class="kw">assume </span><a NAME="E2:3_1_1_1_1"/>
<font color="Maroon" title="c5">b9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> <font color="Maroon" title="c4">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b9</font></span><br/><a NAME="E3:3_1_1_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
( <font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c3">z</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )
 ;<br/><a NAME="E4:3_1_1_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="lattices.html#T16" onmouseover="rs('lattices/T16')" onmouseout="rh()">LATTICES:16</a></span>;<br/><span class="kw">hence </span><a NAME="E5:3_1_1_1_1"/>
<font color="Maroon" title="c4">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="lattice3.html#T7" onmouseover="rs('lattice3/T7')" onmouseout="rh()">LATTICE3:7</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>
<a NAME="E6:3_1_1_1"/>
<font color="Maroon" title="c4">z9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E7:3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A5</font></span>: 
<font color="Maroon" title="c4">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="ref" href="yellow_0.html#T30" onmouseover="rs('yellow_0/T30')" onmouseout="rh()">YELLOW_0:30</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> holds <br/><font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font></span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</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="c6">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> 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="c6">a</font> )</span><br/><span class="kw">assume </span><a NAME="E2:3_1_1_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</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="c6">a</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a9</font> = <font color="Maroon" title="c6">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E4:3_1_1_1_2"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E7:3_1_1_1"><span class="lab"><font color="Green" title="E4">A5</font></span></a>, <a class="ref" href="yellow_0.html#D11" onmouseover="rs('yellow_0/D11')" onmouseout="rh()">YELLOW_0:def 11</a></span>;<br/><a NAME="E5:3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A6</font></span>: 
<font color="Maroon" title="c5">x9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T16" onmouseover="rs('lattices/T16')" onmouseout="rh()">LATTICES:16</a></span>;<br/><a NAME="E6:3_1_1_1_2"/>
( <font color="Maroon" title="c7">a9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a9</font> &amp; <font color="Maroon" title="c5">x9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x9</font> )
 ;<br/><span class="kw">hence </span><a NAME="E7:3_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1_1_1_2"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="lattice3.html#T7" onmouseover="rs('lattice3/T7')" onmouseout="rh()">LATTICE3:7</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">hence </span><a NAME="E9:3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:3_1_1"/>
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D4" onmouseover="rs('yellow_0/D4')" onmouseout="rh()">YELLOW_0:def 4</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>
