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<span class="kw">let </span><font color="Maroon" title="c1">O</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">SubPartialOrdered</a>  iff <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V13" title="OPOSET_1:attr.13">PartialOrdered</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:99"/>
( <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">SubPartialOrdered</a>  implies <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V13" title="OPOSET_1:attr.13">PartialOrdered</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V13" title="OPOSET_1:attr.13">PartialOrdered</a>  implies <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">SubPartialOrdered</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:99_1"/>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">SubPartialOrdered</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V13" title="OPOSET_1:attr.13">PartialOrdered</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">O</font> = <font color="Maroon" title="c1">O</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">SubAntisymmetric</a>   <a href="oposet_1.html#V8" title="OPOSET_1:attr.8">SubTransitive</a>   <a href="qmax_1.html#L2" title="QMAX_1:struct.2">OrthoRelStr</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D24" target="_self" title="OPOSET_1:def.24">Def24</a></span>;<br/>
<a NAME="E3:99_1"/>
( <font color="Maroon" title="c2">O</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">SubAntisymmetric</a>  &amp; <font color="Maroon" title="c2">O</font> is  <a href="oposet_1.html#V8" title="OPOSET_1:attr.8">SubTransitive</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:99_1"/>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V13" title="OPOSET_1:attr.13">PartialOrdered</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D25" target="_self" title="OPOSET_1:def.25">Def25</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">assume </span><a NAME="E2:99"/>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V13" title="OPOSET_1:attr.13">PartialOrdered</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">SubPartialOrdered</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">O</font> = <font color="Maroon" title="c1">O</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="qmax_1.html#L2" title="QMAX_1:struct.2">OrthoRelStr</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D25" target="_self" title="OPOSET_1:def.25">Def25</a></span>;<br/>
<a NAME="E4:99"/>
<font color="Maroon" title="c2">O</font> is  <a href="oposet_1.html#V8" title="OPOSET_1:attr.8">SubTransitive</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E5:99"/>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">SubPartialOrdered</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D24" target="_self" title="OPOSET_1:def.24">Def24</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/>


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