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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</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="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:40"/>
( <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</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="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</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="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a>  implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b2">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Olive" title="b1">x</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D24" onmouseover="rs('waybel_1/D24')" onmouseout="rh()">WAYBEL_1:def 24</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="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D24" onmouseover="rs('waybel_1/D24')" onmouseout="rh()">WAYBEL_1:def 24</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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  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> st <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">y</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="E3:40_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</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="c3">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E4:40_1"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:40_1"/>
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="yellow_7.html#T35" target="_self" onmouseover="rs('yellow_7/T35')" onmouseout="rh()">Th35</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="E2:40"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 for <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"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span>  ex <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"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a></span>)</span> st <font color="Olive" title="b2">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Olive" title="b1">x</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D24" onmouseover="rs('waybel_1/D24')" onmouseout="rh()">WAYBEL_1:def 24</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="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D24" onmouseover="rs('waybel_1/D24')" onmouseout="rh()">WAYBEL_1:def 24</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">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">y</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><br/><a NAME="E4:40"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</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="#E2:40"><span class="lab"><font color="Green" title="E28">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</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">  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E5:40"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:40"/>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="yellow_7.html#T35" target="_self" onmouseover="rs('yellow_7/T35')" onmouseout="rh()">Th35</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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