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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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="c1">L</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  iff <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:37"/>
( <font color="Maroon" title="c1">L</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  implies <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</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="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  implies <font color="Maroon" title="c1">L</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c2">x</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><a NAME="E2:37_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<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 <font color="Maroon" title="c1">L</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D4" onmouseover="rs('yellow_0/D4')" onmouseout="rh()">YELLOW_0:def 4</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="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D5" onmouseover="rs('yellow_0/D5')" onmouseout="rh()">YELLOW_0:def 5</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:37_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="yellow_7.html#T8" target="_self" onmouseover="rs('yellow_7/T8')" onmouseout="rh()">Th8</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">given </span><font color="Maroon" title="c2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span><span class="kw"> such that </span><a NAME="E3:37"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D5" onmouseover="rs('yellow_0/D5')" onmouseout="rh()">YELLOW_0:def 5</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="c1">L</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> </span><br/>

<span class="kw">take </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D4" onmouseover="rs('yellow_0/D4')" onmouseout="rh()">YELLOW_0:def 4</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">  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <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 <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">thus </span><a NAME="E4:37"/>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <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 <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="yellow_7.html#T9" target="_self" onmouseover="rs('yellow_7/T9')" onmouseout="rh()">Th9</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/>


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