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<a NAME="E1:89_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">S</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T56" target="_self" title="YELLOW_0:th.56">Th57</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">S</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="c2">S</font> is  <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:89_1_1"/>
<font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</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="c2">S</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D18" target="_self" title="YELLOW_0:def.18">YELLOW_0:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:89_1_1_1"/>
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 ; <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>

<span class="kw">thus </span><a NAME="E4:89_1_1"/>
<font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D19" target="_self" title="YELLOW_0:def.19">YELLOW_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:89_1_1_2"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 ; <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>

<span class="kw">thus </span><a NAME="E5:89_1_1"/>
(  not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  )
 ; <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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