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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  1 <a href="struct_0.html#V13" title="STRUCT_0:attr.13">-element</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="qmax_1.html#L2" title="QMAX_1:struct.2">OrthoRelStr</a> ; <a class="txt" 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="oposet_1.html#V10" title="OPOSET_1:attr.10">OrderInvolutive</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V7" title="OPOSET_1:attr.7">Pure</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">PartialOrdered</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">L9</font> = <font color="Maroon" title="c1">L</font> as  1 <a href="struct_0.html#V13" title="STRUCT_0:attr.13">-element</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="qmax_1.html#L2" title="QMAX_1:struct.2">OrthoRelStr</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> =  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c2">L9</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:66_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T131" title="ZFMISC_1:th.131">ZFMISC_1:131</a></span>;<br/>
<a NAME="E5:66_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="funct_2.html#T51" title="FUNCT_2:th.51">FUNCT_2:51</a></span>;<br/>
<a NAME="E6:66_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="partit_2.html#V1" title="PARTIT_2:attr.1">involutive</a> 
 
;<br/>
<a NAME="E7:66_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c2">L9</font> is  <a href="oposet_1.html#V1" title="OPOSET_1:attr.1">Dneg</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D3" title="OPOSET_1:def.3">OPOSET_1:def 3</a></span>;<br/>
<a NAME="E8:66_1_1"/>
 for <font color="Olive" title="b1">z</font>, <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="c2">L9</font>  st <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/>

<a NAME="E2:66_1_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="funct_2.html#T50" title="FUNCT_2:th.50">FUNCT_2:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:66_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></span>; <a class="txt" 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="E9:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#D1" title="WAYBEL_9:def.1">WAYBEL_9:def 1</a></span>;<br/>
<a NAME="E10:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> is  <a href="oposet_1.html#V9" title="OPOSET_1:attr.9">Orderinvolutive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="oposet_1.html#D17" title="OPOSET_1:def.17">OPOSET_1:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:66_1_1"/>
( <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V10" title="OPOSET_1:attr.10">OrderInvolutive</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V7" title="OPOSET_1:attr.7">Pure</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">PartialOrdered</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="oposet_1.html#D15" title="OPOSET_1:def.15">OPOSET_1:def 15</a>, <a class="ref" href="oposet_1.html#D18" title="OPOSET_1:def.18">OPOSET_1:def 18</a></span>; <a class="txt" 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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