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

<span class="kw">set </span><font color="Maroon" title="c1">O</font> =  <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">C</font> = the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">S</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ;<br/>
<a NAME="E1:130"/>
the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  <a href="oposet_1.html#R1" title="OPOSET_1:pred.1">QuasiOrthoComplement_on</a>  <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  ;<br/>
<a NAME="E2:130_1"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
for <font color="Olive" title="b1">x</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 <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  holds  <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><a href="funct_5.html#K8" title="FUNCT_5:func.8">op1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <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 <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="funct_5.html#K8" title="FUNCT_5:func.8">op1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E1:130_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="funct_5.html#K8" title="FUNCT_5:func.8">op1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="oposet_1.html#E4"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="oposet_1.html#T3" target="_self" title="OPOSET_1:th.3">Th3</a>, <a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:130_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="funct_5.html#K8" title="FUNCT_5:func.8">op1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:130_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  holds <br/> (  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  )</span><br/>

<a NAME="E1:130_1_2"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:130_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:130_1_2"/>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T38" title="YELLOW_0:th.38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:130_1"/>
the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  <a href="oposet_1.html#R1" title="OPOSET_1:pred.1">QuasiOrthoComplement_on</a>  <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D34" target="_self" title="OPOSET_1:def.34">Def34</a>, <a class="ref" href="oposet_1.html#T51" target="_self" title="OPOSET_1:th.51">Th51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:130"/>
 <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  is  <a href="oposet_1.html#V19" title="OPOSET_1:attr.19">QuasiOrthocomplemented</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D35" target="_self" title="OPOSET_1:def.35">Def35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
