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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> implies ( <font color="Maroon" title="c1">S</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> ) )</span><br/>

<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E29">A1</font></i>: 
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"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></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"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> st <font color="Olive" title="b2">y</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24" title="WAYBEL_1:def.24">WAYBEL_1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a> )</span><br/>

<a NAME="E2:71"/><i><font color="Green" title="E30">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<b>thus </b><a NAME="E3:71"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">T</font> is <a href="waybel_1.html#V10" title="WAYBEL_1:attr.10">complemented</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24" title="WAYBEL_1:def.24">WAYBEL_1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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="c3">s</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span><b> such that </b><br/><a NAME="E3:71_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E1:71"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c3">s</font></span><br/>

<a NAME="E4:71_1"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow10.html#E2:71"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1"/>
<font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="yellow10.html#E3:71_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="yellow10.html#T17" target="_self" title="YELLOW10:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24" title="WAYBEL_1:def.24">WAYBEL_1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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="c3">t</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span><b> such that </b><br/><a NAME="E6:71"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E1:71"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c3">t</font></span><br/>

<a NAME="E7:71"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow10.html#E2:71"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:71"/>
<font color="Maroon" title="c5">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="txt" href="yellow10.html#E6:71"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="yellow10.html#T17" target="_self" title="YELLOW10:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
