<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">O</font> =  <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>
<b>set </b><font color="Maroon">C</font> = the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>
<b>set </b><font color="Maroon">S</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  as   <a href="struct_0.html#NM6">Function</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  ;<br/>
<a NAME="E2:129"/>
<font color="Maroon">f</font> <a href="oposet_1.html#R2" target="_self">OrthoComplement_on</a>  <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<a NAME="E1:129_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  holds  <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><a href="vectsp_2.html#K7">op1</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>

<a NAME="E1:129_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="vectsp_2.html#K7">op1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E2:129_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="vectsp_2.html#K7">op1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">f</font> = the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  as   <a href="struct_0.html#NM6">Function</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  ;<br/>
<a NAME="E3:129_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  holds <br/> (  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>

<a NAME="E1:129_1_2"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E2:129_1_2"/><i><font color="Green">E31</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:129_1_2_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E2:129_1_2_1"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:129_1_2"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="txt" href="oposet_1.html#E46"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<b>hence </b><a NAME="E4:129_1_2"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="oposet_1.html#D18" target="_self">Def18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:129_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  holds <br/> (  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>

<a NAME="E1:129_1_3"/><i><font color="Green">E33</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E2:129_1_3"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E3:129_1_3"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="oposet_1.html#D18" target="_self">Def18</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:129_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  holds <br/> ( <font color="Olive">x</font> <a href="waybel_1.html#R4">is_maximum_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Olive">x</font> <a href="waybel_1.html#R3">is_minimum_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>

<a NAME="E1:129_1_4"/><i><font color="Green">E35</font></i>: 
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  )
 
<b>by </b><i><a class="ref" href="oposet_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E2:129_1_4"/><i><font color="Green">E120</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E46"><i><font color="Green">Lemma72</font></i></a></i>;<br/>
<a NAME="E3:129_1_4"/>
(  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>, <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  )
 
<b>by </b><i><a class="ref" href="oposet_1.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E4:129_1_4"/>
( <font color="Maroon">a</font> <a href="waybel_1.html#R4">is_maximum_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">a</font> <a href="waybel_1.html#R3">is_minimum_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="oposet_1.html#T30" target="_self">Th30</a>, <a class="ref" href="oposet_1.html#T31" target="_self">Th31</a>, <a class="ref" href="waybel_1.html#D6">WAYBEL_1:def 6</a>, <a class="ref" href="waybel_1.html#D7">WAYBEL_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:129_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  holds <br/> (  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="waybel_1.html#R4">is_maximum_of</a> the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="waybel_1.html#R3">is_minimum_of</a> the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>

<a NAME="E1:129_1_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">a</font> <a href="waybel_1.html#R4">is_maximum_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E2:129_1_5"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</font> <a href="waybel_1.html#R3">is_minimum_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D18" target="_self">Def18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a0</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  ;<br/>
<a NAME="E4:129_1_5"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a0</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a0</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E5:129_1_5"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a0</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a0</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:129_1_5"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="waybel_1.html#R4">is_maximum_of</a> the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="waybel_1.html#R3">is_minimum_of</a> the <a href="struct_0.html#U1">carrier</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  )
 <b>by </b><i><a class="ref" href="oposet_1.html#T28" target="_self">Th28</a>, <a class="ref" href="oposet_1.html#T30" target="_self">Th30</a>, <a class="ref" href="oposet_1.html#T31" target="_self">Th31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:129_1"/>
<font color="Maroon">f</font> <a href="oposet_1.html#R2" target="_self">OrthoComplement_on</a>  <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> 
 <b>by </b><i><a class="ref" href="oposet_1.html#T28" target="_self">Th28</a>, <a class="ref" href="oposet_1.html#D13" target="_self">Def13</a>, <a class="ref" href="oposet_1.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:129"/>
 <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  is <a href="oposet_1.html#V21" target="_self">Orthocomplemented</a>
 <b>by </b><i><a class="ref" href="oposet_1.html#T26" target="_self">Th26</a></i>;<br/>


</div>
